The Osmotic Pressure of Concentrated Protein Solutions Effect of
`Concentration and pH in Saline Solutions of
`Bovine Serum Albumin
`VINCENT L VILKER1 CLARK K COLTON AND KENNETH A SMITH
`
`Department of Chemical Engineering Massachusetts
`
`Institute of Technology Cambridge Massachusetts 02139
`
`Received April 17 1980 accepted July 18 1980
`
`are reported as a function of bovine serum albumin BSA
`Osmotic pressure measurements
`in 015 M sodium chloride at pH 45 54 and 74 The measured values increased
`concentration
`and with increasing pH and therefore
`markedly with increasing BSA concentration
`of 450 gliter solution and a pH of 74 osmotic pressure was
`macroion charge At a concentration
`the same con
`nearly five atmospheres which is more than four times the value measured at
`centration and a pH of 45 and about 30 times the value expected for an ideal solution A semi
`empirical analytical expression was developed which gave good agreement between prediction
`and the experimental data of this and other studies The data were also compared to the prediction
`of a three term virial equation wherein the second and third virial
`coefficients were calculated
`by using McMillanMayer solution theory The expression for the potential of mean force was
`obtained by comparing various contributions to the potential energy of interaction The terms for
`repulsion and dispersion attraction are the same as those used in the DLVO theory
`electrostatic
`of colloid stability The predicted curves are of the correct order of magnitude and follow the cor
`rect qualitative trend with pH but
`to display the strong pH dependence of the data The
`they fail
`factors responsible for this deficiency are assessed and opportunities
`for developing a more realistic
`
`increasing
`
`potential
`
`function are identified
`
`INTRODUCTION
`
`When a protein solution is ultrafiltered
`by a membrane a region of increased con
`centration of the retained solute develops
`near the membrane surface The concentra
`tion at the surface can approach or even at
`tain the solubility limit for the protein and
`the driving force for hydraulic flow is re
`duced by the increased osmotic pressure dif
`the membrane This
`ference
`across
`phenomenon of concentration polarization
`can thus greatly reduce the hydraulic flux
`as compared to that attainable with pure
`water
`In order to obtain a fundamental
`understanding of protein ultrafiltration data
`
`1 Author to whom correspondence
`should be sent
`Chemical Nuclear and Thermal Engineering Depart
`ment University of California Los Angeles California
`90024
`
`are required for the transport and osmotic
`pressure properties of these concentrated
`solutions
`In the past osmotic pressure measure
`ments of protein solutions have generally
`been confined to the dilute range and have
`been taken primarily for the purpose of ob
`taining molecular weight and conforma
`tional data 13 In only a few instances
`eg 48 have measurements been made
`up to moderate concentrations nor are
`existing theoretical models of highly non
`ideal solution behavior suitable for a priori
`prediction at high concentration The tradi
`tional approach is the Donnan membrane
`equilibrium model Within this context
`the
`exact multicomponent chemical potential
`treatment of Scatchard 4 5 simply cor
`relates data within the range for which it
`is
`available The same is true for semiquantita
`
`002197978102054819$02000
`Copyright g 1981 by Academic Press Inc
`All rights of reproduction in any form reserved
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`548
`
`Abraxis EX2044
`Actavis LLC v Abraxis Bioscience LLC
`1PR201701101 1PR201701103 1PR201701104
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`549
`
`ranging
`
`tive interpretation of the osmotic virial coef
`ficients of protein solutions in terms of ex
`cluded volume and attractive interaction ef
`
`fects 7 The most promising approach is
`theory 9
`the McMillan Mayer solution
`from which osmotic virial coefficients
`can
`be estimated in a manner analogous to those
`for the pressure of an imperfect gas Hill
`10 11 12 has applied this theory to
`colloid particles which
`exhibit
`charged
`double layer repulsion but no comparison
`with experimental data has heretofore been
`attempted
`In this paper we report osmotic pressure
`measurements for solutions of bovine serum
`albumin BSA at concentrations
`from 84 up to 475 gliter solution in 015 M
`sodium chloride at pH 45 54 and 74 The
`measurements were made with a static
`membrane osmometer built to withstand the
`several atmospheres of pressure generated
`by these solutions The data are fit by a semi
`empirical correlation suggested by Donnan
`theory that also gives good agreement with
`data from other studies Lastly the con
`tributions to the potential energy of inter
`action between albumin molecules in solu
`tion are evaluated using physical properties
`available in the literature and the result
`ing expression for the potential of mean
`force is used with the McMillan Mayer
`theory to predict second and third osmotic
`The poor agreement
`virial coefficients
`that
`results between predicted and measured os
`motic pressure reflects the inadequacy of a
`three term virial expansion at
`the higher
`examined and it
`concentrations
`protein
`highlights the need for a better description
`of the potential of mean force than is cur
`rently available to describe the strong pH
`of the data
`dependence
`
`MATERIALS AND METHODS
`
`Albumin
`Albumin solution
`solutions
`were prepared by mixing BSA crystals
`Pentex grade recrystallized Cohn Frac
`tion IV cat no 81001 Miles Laboratories
`Kanakkee Illinois with 015 M NaC1 made
`
`from distilled water and analytical grade
`NaCl All prepartions included
`sodium
`azide ca 10 mgliter as an antibacterial
`above about 300
`agent For concentrations
`solution BSA crystals
`and
`saline
`gliter
`were added to 50ml centrifuge tubes which
`were agitated by vigorous vortexing motion
`Albumin crystals were used as received
`According
`to the manufacturer
`the final
`ion
`before
`recrystallization were
`steps
`exchange which ideally removed all micro
`
`ions except W and OW followed by ad
`dition of NaOH to raise the pH to 52 The
`average chloride ion content was 3 mgg pro
`tein No special steps were taken to remove
`bound lipids Cellulose acetate electro
`phoresis in this study indicated 100 al
`bumin purity and acrylamide gel electro
`showed 47 polymer
`bands
`phoresis
`thereby indicating the presence of some al
`bumin oligomers
`Solution pH measurements ±001 pH
`unit were made with a saturated KC1 glass
`electrode Solution adjustment of pH was
`01 N
`made by addition of nonbuffered
`NaOH or HC1 Vigorous vortex mixing was
`local protein de
`employed to ensure that
`naturation would not occur during acid or
`The
`base addition
`solutions were not
`analyzed for sodium or chloride ion con
`centrations Because of the large aliquots of
`01 N NaOH or 01 N HC1 which were
`added for pH adjustment and the slight
`variability of a content of different lots of
`
`albumin crystals the concentrations of Nat
`and Cl following pH adjustment were
`from the 015 M saline
`slightly different
`added The maximum difference
`initially
`solution is es
`for the most concentrated
`timated to be about 003 M
`
`All albumin solutions were noncloudy
`but occasionally small strands of apparently
`denatured protein were observed For this
`reason the final step before an experimental
`run was filtration through a 01µm filter
`for
`albumin concentration
`up to 300 gliter or a
`03µm filter
`solutions of higher
`concentration
`
`for
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`

`

`550
`
`VILKER COLTON AND SMITH
`
`VOLUMETRIC
`C API LLARIES
`RESOLUTION
`= 0002 MO
`
`SOLUTIO Fl
`25 MU
`
`PRECISION
`PRESSURE
`GAGE
`
`PRECISION
`PRESSURE
`CONTROLLER
`
`CONSTANT
`TEMPERATURE
`COIL
`
`PRESSURIZED
`NITROGEN
`CYLINDER
`
`SOLVENT
`25 ML
`
`MEMBRANE
`
`FIG 1 Schematic diagram of highpressure mem
`brane osmometer system
`
`Albumin solutions charged to and dis
`charged from the osmometer were analyzed
`for pH and for albumin concentration with
`the biuret method 13 The solution dis
`charged from the solvent chamber was also
`checked
`for possible albumin
`routinely
`leakage with the biuret method or the Lowry
`method 14 which is more accurate when
`are very low The
`protein concentrations
`precision of concentration measurement
`
`was ±5
`
`pressure measurement
`Osmotic
`osmotic pressure measurement
`system is
`shown schematically in Fig 1 The os
`mometer cell consists of
`two chambers
`
`The
`
`one for the 015 M saline solvent and one
`for the albumin solution The
`chambers
`are separated by a membrane which is im
`permeable to albumin but permits free pas
`and microions After
`sage of water
`the
`chambers
`a volumetric capil
`are filled
`lary prefilled with the appropriate solution
`is connected to each chamber The gas pres
`sure applied to the capillary on the solu
`to the es
`tion chamber is then quickly set
`timated osmotic pressure and subsequently
`adjusted in the direction indicated by slight
`in capillary liquid levels Ul
`movements
`timately an applied pressure is found for
`which liquid levels do not change over a
`period of several hours This pressure is
`taken to be the solution colloid osmotic
`pressure Applied pressure is measured and
`controlled to within several mm Hg by use
`of a precision pressure regulation gauge
`The resolution of volume flow measurement
`by the volumetric capillaries is about 0002
`ml The osmometer cell and gas temperature
`equilibration coil are immersed in a tem
`perature bath controlled at 25 ± 01°C
`the os
`Figure 2 is a detailed view of
`mometer cell The chambers are formed by
`a membrane between
`two
`sandwiching
`pieces of Plexiglas each of
`
`cylindrical
`
`REMOVABLE
`CAPILLARY
`
`MEMBRANE
`
`AREA 99 CM2
`
`ULMER
`0 RING SEAL
`
`FILLING PORT
`
`SOLVENT
`
`CHAMBER
`
`RIGID
`
`SCREEN
`
`POROUS
`
`FRIT SUPPORT
`
`SOLUTION
`DEPTH
`
`CHAMBER
`025 cm
`
`DISCMGRGE PORT
`PLUG NOT SHOWN
`
`FIG 2 Exploded
`
`view of the membrane osmometer cell
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`551
`
`4
`
`301
`
`20
`
`0
`
`10
`
`201
`
`40
`
`50
`
`60
`
`70
`
`PH
`
`serum albumin charge Z bound
`FIG 3 Bovine
`ions vie and bound chloride ions Pa
`hydrogen
`per albumin molecule in 015 M NaC1 solution as a
`function of solution pH Isoelectric
`pH = 472 iso
`ionic pH = 546
`
`which contains a shallow circular depres
`sion 025 cm deep x 356 cm diameter
`The membrane is supported on the solution
`side by a metal screen
`and a porous frit
`One eighth in diameter stainless steel rods
`not shown equally
`around the
`spaced
`chambers about halfway to the outer perim
`eter are used to clamp the unit
`together
`A rubber 0 ring impressed on the solvent
`the membrane seals
`side of
`the unit
`to
`least 4500 mm Hg
`applied pressures of at
`when the two halves are clamped The cel
`lulosic membranes Abcor HFA180 sheet
`stock used for all determinations
`have a
`rejection coefficient of 099+ for albumin
`and not more than 2 x 104 for saline 15
`Five membranes were used in the course of
`about 50 experimental measurements with
`no detectable
`in results for
`differences
`different membranes
`At the conclusion of each measurement
`solvent and solution samples are taken with
`a syringe and needle via the filling ports
`For concentrations
`of about 400 gliter or
`more rapid sample discharge was attained
`by removing the plug to the discharge port
`with the solution under pressure
`To confirm that stable liquid levels in the
`
`of true thermo
`capillaries are indicative
`separate de
`equilibrium two
`dynamic
`terminations were made on identical start
`ing solutions In one the initial
`applied
`pressure was less than the osmotic pres
`sure of the solution in the second it was
`greater In each case the applied pressure
`was adjusted until volume transfer between
`and the two osmotic
`ceased
`chambers
`pressure measurements
`
`about 4 Additional details are available
`elsewhere 15
`
`agreed to within
`
`Albumin valence calculation For use in
`the models subsequently employed in this
`paper the average net molecular charge of
`albumin is
`from its
`complex
`calculated
`equilibria with 11+ and Cl ions In the pH
`range of our experiments Na+ binding is un
`important 16 and the availability of bind
`ing sites for H+ and Cl is constant since
`in the protein second
`there are no changes
`ary structure 17 The macroion charge
`number Z is equal to the difference between
`the number of bound protons vii+ and the
`bound
`vcr per albumin
`chloride ions
`
`molecule
`
`vct
`
`1
`Z = PH+
`The isoelectric pH Z = 0 in 015 M saline
`solutions is about 472 18 19 The average
`albumin charge number is obtained by com
`bining Tanfords proton binding data from
`titration measurements in 015 M NaCl 17
`with the two site chloride binding model of
`Scatchard et al 20
`nkiCly exp2wZ
`1 + kiClly exp2wZ
`n2k2Cly exp2wZ
`1 + k2Cly exp2wZ
`where n = 10 lc = 44M1 n2 = 30 k2
`= 11 M1 and Cl is the free chloride ion
`in solution 015 M The
`concentration
`parameters y and w are calculated for our
`conditions to be 078 and 0026 respectively
`15 For a given pH vii+ is found from
`Tanfords titration data as shown in Fig 3
`
`2
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`

`

`552
`
`VILKER COLTON AND SMITH
`
`is then used to
`
`and iterative calculation
`solve Eqs 1 and 2 simultaneously for the
`values of vcr and Z These results are also
`shown in Fig 3
`The isoionic pH vH+ = 0 by definition
`measured following addition of BSA 50 g
`liter to 015 M NaC1 ranged from 522 to
`555 pH for the various lots of albumin used
`in this work These values are in good agree
`ment with 546 pH as given by Tanford 17
`EXPERIMENTAL
`RESULTS
`The albumin concentration C and pH
`measured in the solution discharged from
`the osmometer
`albumin
`the calculated
`charge number and the measured osmotic
`pressure are tabulated in Table I The dis
`charge concentration
`varied from the initial
`concentration by ±10 at most and the pH
`of the discharged solution was never sig
`nificantly different ± 005 pH units from its
`initial value The albumin concentration
`in
`the solvent chamber discharge was usually 1
`to 3 gliterThese low concentrations did not
`to the re
`contribute significant corrections
`ported osmotic pressures From tests for
`thermodynamic equilibrium the precision
`of osmotic pressure measurements was es
`timated to be within ±5
`Reduced osmotic pressure 7rIC0 is plotted
`in Fig 4 as a function of albumin concen
`tration The data at the lowest concentration
`for each pH are consistent with extrapola
`tion to a value for the intercept RTIM
`= 0270 mm Hggliter solution which cor
`responds to the molecular weight of 69000
`first determined by Scatchard 5 This value
`
`acid
`
`sequencing
`from the
`
`than that for monomeric albumin
`is higher
`determined by amino
`66100 21 and it could result
`presence of about 5 dimers or higher
`
`oligomers The osmotic pressure exhibits a
`strong dependence on albumin concentra
`tion and solution pH At all values of pH
`the slope at low concentration and conse
`quently the second virial
`coefficient
`positive The nonlinear increase in 711C0
`with increasing C indicates that
`the third
`
`is
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`TABLE I
`
`Osmotic Pressure of Bovine Serum Albumin Solutions
`
`Albumin con
`centration Cv
`
`gliter solution
`
`Solution
`pH
`
`Albumin
`charge Z
`
`Osmotic
`
`pressure sr
`
`mm Hg
`
`84
`
`91
`
`211
`
`211
`289
`
`325
`
`325
`
`354
`
`357
`
`413
`
`428
`
`448
`
`91
`
`130
`
`144
`
`234
`
`240
`
`245
`
`338
`
`395
`
`411
`
`414
`
`430
`
`454
`
`126
`
`182
`
`278
`
`317
`
`318
`
`343
`
`418
`
`475
`
`735
`737
`740
`746
`748
`734
`738
`740
`750
`744
`744
`742
`
`541
`540
`540
`544
`545
`542
`540
`542
`544
`542
`541
`
`545
`
`446
`454
`452
`450
`450
`452
`457
`450
`
`202
`203
`204
`206
`207
`202
`203
`204
`208
`206
`206
`205
`
`92
`91
`91
`95
`96
`93
`91
`93
`95
`93
`92
`96
`+55
`+36
`+41
`+45
`+45
`+41
`+32
`+45
`
`48
`
`59
`
`332
`
`334
`
`844
`
`996
`
`996
`
`1423
`
`1638
`2620
`
`2806
`
`3640
`
`41
`
`74
`
`90
`
`260
`
`229
`
`269
`
`618
`
`1005
`
`1230
`
`1286
`
`1370
`
`1529
`
`47
`
`93
`
`182
`
`228
`
`244
`
`284
`
`716
`
`889
`
`the virial expansion makes a sig
`term of
`nificant contribution to the osmotic pres
`the lowest concentrations
`sure at all but
`studied At
`concentration
`highest
`the
`examined the osmotic pressure about
`the pH 74 solution is about
`atm of
`five
`times larger than that of the pH 45 solution
`and about 30 times larger than the value
`predicted for an ideal solution by the vant
`Hoff equation
`In order to describe these results by an
`analytical expression we fit
`the albumin
`osmotic pressure data to the following semi
`
`5
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`553
`
`A = 2950 x 105
`
`A2
`
`5625 x 104
`
`taken the coefficients of the second and third
`terms to be quadratic functions of charge
`A2 and A were evaluated by nonlinear least
`squares regression analysis to yield
`2410 x 104Z
`3664 x 105Z2 4
`1051 X 106Z
`+ 1762 x 107Z2 5
`The prediction of Eqs 3 to 5 is shown
`by the curves in Fig 4 which give a good fit
`to the experimental data
`In Fig 5 the semiempirical correlation is
`compared with osmotic pressure data for
`low and moderate albumin concentrations
`from Scatchard et al 5 and Kappos and
`Pauly 8 The correlation predicts slightly
`higher values than those measured at 63 pH
`
`20
`
`15
`
`REF
`
`0
`
`5
`
`pH
`
`54
`
`A 5 63
`8
`
`63
`
`No CI
`
`conc m
`
`CURVE
`
`015
`07
`
`070
`
`10In
`
`05
`
`cn
`
`k
`
`cco
`
`0 1
`
`7 r
`
`e
`
`01
`0
`
`00
`
`200
`
`300
`
`ALBUMIN
`
`CONCENTRATION
`
`Cp
`
`g
`
`I SOLUTION
`
`FIG 5 Comparison of BSA osmotic pressure data
`by semi
`from other studies with curves predicted
`empirical correlations of Eqs 3 to 5
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`pH
`
`74
`
`54
`
`45
`
`A
`
`0
`
`0
`
`90
`
`80
`
`Z 70
`m
`
`E 60
`
`50
`
`E a
`
`k l
`
`ii
`cc
`
`m 40
`
`0L
`
`a
`
`30
`
`20
`
`101
`
`cco
`
`0 0 a
`
`11
`CC
`
`°0
`
`60
`
`260
`
`300
`
`ALBUMIN CONCENTRATION
`
`460
`500
`Cp VP SOLUTION
`
`FIG 4 BSA reduced osmotic pressure as a function
`at 25°C and in 015 M
`of albumin
`concentration
`at pH 74 54 and 45 Curves
`from Eqs 3 to 5
`
`are derived
`
`NaC1
`
`12
`
`in
`
`empirical function of albumin concentration
`ZC 2
`71 RTI2 +mai 2m0
`2M
`+ RT C + i120 + A3G 3
`M
`where 10 is protein molecular weight
`is molar salt concentration and C is al
`bumin concentration gliter solution The
`term in braces accounts for the ideal
`first
`Donnan
`and the
`second
`term
`effect
`accounts for nonidealities arising from inter
`actions of albumin macroions microions
`and water between themselves
`and each
`other This approach was
`by
`suggested
`based upon Donnan
`previous treatments
`Theory 4 5 22 23 We have further ex
`pressed the second term in the form of a
`term of which is
`virial expansion the first
`the ideal vant Hoff contribution and have
`
`

`

`554
`
`VILKER COLTON AND SMITH
`and 017 M NaCl Otherwise the agreement
`
`where
`
`with these other studies is excellent
`
`THEORY
`Following the development of Hill 10
`12 we consider
`a system in which the
`chemical potentials of water and microions
`in both solutions while albumin
`are equal
`is present on only one side of the semi
`permeable membrane The osmotic pres
`sure can be equated to a virial expansion
`in powers of the solute number density
`
`IT = c + B2c2 + B3c3 +
`kT
`
`6
`
`where the virial coefficients Bn have dimen
`sions cm3moleculen1 The solute number
`density is related to the weight concentra
`tion by
`
`C2r12 =
`
`f13f23dr3
`
`11
`
`The B3 triple integral
`is partitioned to the
`form of Eq 10 as suggested
`by Barker
`and Henderson 24 in order to facilitate
`the numerical computation
`discussed later
`in this paper The analogy inherent
`in this
`between the two component
`development
`system of solute in solvent and that of
`gas in vacuum is valid for solutions in
`which intermolecular
`are of suf
`forces
`ficiently short range to ensure convergence
`of the cluster
`integrals This constraint
`satisfied for dilute solutions of macroions
`which contain sufficient electrolyte to pro
`vide Debyetype screening of the coulombic
`interactions
`
`is
`
`7
`
`Intermolecular Interactions
`
`In order
`
`to construct
`
`a pair potential
`
`function various contributions to the inter
`between
`albumin molecules
`action
`in
`electrolyte solution are examined Analyt
`ical expressions
`for these contributions
`applicable to a spherical molecule of radius
`a are summarized in Table II
`Each of the electrostatic
`interactions
`expressed as a product
`
`is
`
`W = W
`12
`W is the pair potential function for particles
`
`immersed in a medium of dielectric constant
`E but without ionic double layers it reduces
`to the expression for interaction between
`ions in a vacuum when E = 126 The factor
`is a number less than unity which ac
`counts for the screening of these interac
`tions by the particle double layers Similar
`screening effects occur with the induction
`interactions but screening factors for these
`cases are not available in the literature
`for charge dipole
`dipole dipole and dipole induced dipole
`are the effective
`interactions
`spherically
`averaged
`
`The W expressions
`
`symmetric potential
`
`functions
`
`c
`
`NAC
`103M
`to McMillan Mayer
`
`According
`
`theory
`
`9 the virial coefficients can be expressed
`
`in terms of cluster
`
`integrals and the
`
`function
`
`8
`= expWukT
`where Wu subsequently denoted by W is
`
`1
`
`between
`
`the potential of average force
`solute albumin pairs i and j
`in an infinitely
`dilute solution c > 0 with center to
`center separation r which we approximate
`by the intermolecular potential function If
`the potentials are spherically symmetric the
`two coefficients in Eq 6 are given by
`first
`
`B2 =
`
`1
`
`2V 11v f12dr1dr2
`
`= 27r
`
`f12r2dr
`
`9
`
`1
`
`f
`
`3 V iv Iv Iv
`
`f12f1323dr1dr2dr3
`
`=
`
`47r
`
`3
`
`10
`
`C2rA2r2dr
`
`10
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`555
`
`TABLE II
`
`Contribution to the Intermolecular Potential Function
`
`Desig
`
`nation
`
`Unscreened
`
`potential function 14
`
`Screening factor
`
`Type
`
`Electrostatic
`
`Charge charge
`
`Wqq
`
`Charge dipole
`
`W5
`
`Dipole dipole W`
`
`Charge fluctua
`
`tion
`
`Induction
`
`Charge induced
`
`W40
`
`dipole
`
`Dipole induced
`dipole
`
`Dispersion
`
`W
`
`W aor
`
`+ Ze2
`
`Er
`
`2 Ze2µ2
`
`3
`
`E2kTr4
`
`2
`
`3
`
`10
`
`kTE2r6
`
`Z22e4
`
`2e2kTr2
`
`Ze2a
`
`62r4
`
`2 tea
`3 E2r6
`
`A 2
`6s2
`
`where
`
`2
`
`s2
`
`4
`
`+ ln
`
`s2
`
`4
`
`s
`
`A = 42 AI22
`Ai 72PiN3 h v0 a2
`
`4
`
`Mi
`
`S = rla
`
`geidn2a
`1 + Ka2
`31 I Krer2
`1 + Ka2 + 2Ka + Ka2
`+ 1 + KaEsIE
`32 + 2xr + Kr22e0 r
`12 + 2Ka + 1a2 + 1 + Ka€€
`
`2
`
`1
`
`e2dr2¢
`1 + 2Ka2
`
`Unknown
`
`Unknown
`
`i = p s
`
`Refer
`
`ence
`
`25
`
`26 27
`
`27 28
`
`29
`
`26
`
`28
`
`30 31
`
`over all orientations all other contributions
`tabulated are orientation independent
`Within the domain of the molecule Wi
`= 00 For otherwise
`noninteracting
`rigid
`such that Wu = 0 for
`spheres of radius a
`r > 2a Eqs 9 to 11 reduce to 26
`
`Bg 7a= 4v
`
`16r
`
`3
`
`13
`
`14
`
`B3 = 5Bi
`8
`
`Consequently
`the virial
`coefficients
`are
`usually evaluated by beginning the integra
`tion of Eqs 9 to 11 at
`r = 2a and
`adding the result to the respective excluded
`volume contribution The prolate ellipsoid
`of axial ratio p = alb is a better model of
`albumin for which case the excluded vol
`ume contributions are given by 24
`B2 = Vm
`
`R1S1
`
`15
`
`B3 = 14 ± 2R1S1vm + R1S22
`
`1
`
`3
`
`16
`
`17
`
`and higher order coefficients are given by
`B4 = 028693g
`
`14a
`
`where
`
`B5 = 0115B1
`
`14b
`
`a
`
`R1 = 1 +
`
`2
`
`1
`
`E2
`
`1 ±
`
`In
`
`2€
`
`1
`
`E
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`

`

`556
`
`VILKER COLTON AND SMITH
`
`Si = In b2
`
`1 +
`
`sin1 E
`
`where K is defined by
`
`18
`
`1
`
`p2
`
`19
`
`The dispersion contribution W between
`
`spherical particles was derived by Hamaker
`30 by assuming pairwise additivity of the
`intermolecular
`interaction Far
`from the
`
`particle surface s 1 and in the absence
`of an intervening solvent
`Table II
`reduces to
`
`the expression in
`
`Weea
`
`3 hvcp4
`
`r6
`
`20
`
`4
`which was derived by London 32 for the
`dispersion interaction between two point
`molecules in vacua The dispersion interac
`tion is unaffected
`by ionic double layer
`screening because
`the correlation time of
`the electronic
`between atoms
`fluctuations
`is much smaller than the time for adjustment
`of ions in the double layer
`
`The contribution W originally sug
`
`gested by Kirkwood and Shumaker 29
`arises from time correlation between fluc
`in charge and charge distribution
`tuations
`in number and
`associated with fluctuations
`bound
`configuration of
`albumin
`
`the protons
`
`to
`
`calculation
`
`of
`
`Rigorous
`the repulsive
`charge charge potential energy of interac
`requires numerical solution of the
`tion first
`nonlinear Poisson Boltzmann equation for
`potential distribution sur
`the
`electrical
`rounding a single macroion 3336 When
`the electrical potential qi at the surface of a
`macroion is less than about 50 mV and the
`Debye length K1 is greater
`than about
`one fifth of the macroion radius as is the
`case in this study the description of the
`potential distribution in the double layer by
`the linearized DebyeHiickel
`equation is
`applicable and the electrical potential
`given by
`
`is
`
`Ze
`Er1 + Ka
`
`21
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`K2
`
`4n e2
`
`EkT
`
`E ciZf
`
`22
`
`is
`
`and ci
`
`the ith
`
`is the number density of
`microion of charge number Z The expres
`sion for Wgl
`in Table II
`is based upon these
`approximations and is the relation originally
`developed by Verwey and Overbeek 25
`The parameter g in the screening factor
`less than unity its magnitude depends
`on
`Kr and on the boundary condition employed
`constant surface potential
`lbcp or constant
`For
`surface
`the
`ionic
`charge
`density
`strength and pH applicable here g > 09
`under all conditions 25 Table XX and its
`value was set equal
`to unity for numerical
`evaluation
`An additional repulsive contribution not
`shown in Table II is present at separations
`for which electron clouds begin to overlap
`and nuclei of surface atoms begin to repel
`is ap
`each other This
`contribution
`proximated by W > co and it
`limits the
`
`gliter
`
`separation
`
`particle
`
`surfaces to
`
`between
`some minimum value u The integrations of
`Eqs 9 to 11 are thus begun at r = 2a + if
`and the contributions from the region 2a
`< r
`2a + a are properly included within
`the excluded volume contributions
`The physical parameter estimates used
`evaluation of the potential
`for quantitative
`function are summarized in Table III The
`hydrated density we measured 15 was in
`dependent of albumin concentration to 560
`solution and agrees well with other
`data 1 Estimates of albumin size and
`shape vary widely in the literature 1 37 38
`4244 with molecular volume ranging
`from 86000 42 to 200000 A3 and aspect
`ratio alb ranging from 10 43 to 65 1 The
`estimates of Wright and Thompson 37
`from rotational diffusion measurements in
`low salt solutions at 76 pH were selected as
`being the most reliable and representative
`of recent estimates and they were used
`for calculating equivalent sphere and prolate
`ellipsoid dimensions The tabulated dipole
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`557
`
`TABLE III
`
`Parameters Used to Evaluate Potential Functions
`
`Parameter
`
`Value
`
`Reference
`
`Albumin
`Molecular weight Mi
`Hydrated density p
`Molecular volume v
`radius a
`
`Equivalent
`
`spherical
`
`Ellipsoid shape
`
`Semimajor axis 64
`Semiminor axis °b
`Dipole moment µ
`Polarizability a
`frequency Pop
`Characteristic
`Charge number Z
`
`Root mean square charge number
`
`fluctuation Z212
`
`Characteristic
`
`Solvent water
`Polarizability a
`frequency Po
`Dielectric constant
`Bulk value E
`Local value at macroion surface
`
`Calculated
`
`Debye length ic1
`Albumin surface potential
`
`EN
`
`Hamaker constants
`
`A pA
`
`A
`
`15
`37
`
`37
`
`38
`39
`39
`Fig 3
`
`40
`
`26
`26
`
`41
`27
`
`Eq 22
`Eq 21
`
`69000
`134 gcm3
`128000 A3
`313 A
`
`705 A
`208 A
`380 x 10 esucm
`5950 A3
`306 x 10 sec
`204 at 74 pH
`91 at 54 pH
`+45 at 45 pH
`35
`
`148
`435 x 1015 sec
`
`783
`
`4
`
`78 A
`235 mV at 74 pH
`105 mV at 54 pH
`+52 mV at 45 pH
`
`727 x 10 ergs
`525 x 10 ergs
`165 x 1014 ergs
`
`moment was determined with albumin in
`salt free isoionic solution 38 and albumin
`polarizability and characteristic
`frequency
`was determined by refractive index meas
`urements in 015 M NaC1 39 The charge
`fluctuation was evaluated in salt free iso
`ionic solution 40 A value lower by a factor
`of more than three has also been reported
`45 The Debye length calculated from Eq
`22 is for the 015 M NaCl
`ionic strength
`of our albumin solutions Others 46 have
`found that albumin solution conductance
`measurements are better fit by double layer
`theory when the protein concentration
`is
`included as a 11 electrolyte in Eq 22 This
`
`dependence which
`protein concentration
`would have reduced K1 to 57 A at the high
`concentration
`and albumin charge
`est
`included in our calcula
`studied was not
`tions The
`estimated minimum surface
`separation is 30 A which corresponds
`to
`the minimum separation between peptide
`chains in proteins 47
`
`Pair Potential Function and Osmotic
`Pressure Calculation
`
`various
`
`to the un
`The
`screened e = 1 pair potential energy of
`interaction from Table II are compared in
`
`contributions
`
`Journal of Colloid and Interface Science
`
`Vol 79 No 2 February 1981
`
`

`

`558
`
`VILKER COLTON AND SMITH
`
`a
`
`cc
`
`L
`
`a
`
`$E
`
`16
`
`3
`
`4
`
`5
`
`6
`
`2
`
`3
`
`4
`
`5
`
`6
`
`2
`10
`
`10°
`
`2
`10
`
`164
`
`2
`
`81
`
`0
`
`16
`kT
`
`DIMENSIONLESS
`
`DISTANCE
`
`S r a
`FIG 6 Magnitude of unscreened pair potential energies of interaction for albumin as a function of
`center to center separation distance Solid curves were calculated from equations in Table II Dashed
`curve is from Eq 20 for dispersion interaction between point albumin molecules modified as in Table
`for presence of intervening solvent A pH 74 B pH 45
`to account
`
`II
`
`Fig 6 The potential energy is normalized
`by kT and center to center distance is nor
`malized by the albumin equivalent spherical
`radius a In the graph for pH 45 the same
`contributions which apply at pH 74 are
`present but only those which depend on Z
`have been plotted The largest contribution
`at each pH is the repulsive charge charge
`interaction Charge fluctuations and
`
`chargedipole
`
`are the most important
`interactions
`attractive contributions and the dispersion
`contribution calculated from the Hamaker
`
`equation is important only at very small
`separation
`Figure 7 is a similar plot which includes
`the effects of double layer screening on the
`contributions The magnitude
`electrostatic
`
`attractive com
`of each of the electrostatic
`ponents is substantially reduced Although
`screening factors are not available for the
`
`in
`
`that the
`
`induction contributions
`the results
`Fig 6 provide no reason to expect
`would
`screened
`induction contributions
`be significant The largest contributions are
`chargecharge
`interactions
`the repulsive
`and the attractive
`dispersion interactions
`By ignoring the other contributions the pair
`potential function becomes
`
`W =wqu
`
`wax
`
`23
`
`The interplay between these two contribu
`tions forms the basis for the classical DLVO
`theory 25 48 of colloid stability
`
`FIG 7 Magnitude of screened pair potential energies of interactions A pH 74 B pH 45
`
`DIMENSIONLESS
`
`DISTANCE
`
`S
`
`rid
`
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`2
`
`3
`
`4
`
`5
`
`6
`
`

`

`OSMOTIC PRESSURE OF PROTEIN SOLUTIONS
`
`559
`
`pH 74
`
`pH 54
`
`pH 45
`
`21
`
`22
`
`23
`
`24
`
`25
`
`26
`
`27
`
`28
`
`10
`
`08
`
`06
`
`+OA
`
`02
`
`001
`
`kw1
`
`02
`
`04
`
`If
`
`1
`
`1111
`
`08
`
`10
`20
`
`S ra
`FIG 8 Pair potential energy function calculated from Eq 23 for pH 45 54 and 74 Dashed curves
`functions which would result if repulsive barrier at s = 2 + crla were ignored
`show potential
`
`DIMENSIONLESS
`
`DISTANCE
`
`The pair potential energy functions cal
`culated from Eq 23 are shown in Fig 8
`The variation with pH reflects the change in
`albumin charge Fig 3 Repulsion domi
`nates at pH 74 and the maximum value of
`WIkT is about one Repulsion dominates at
`pH 54 and attraction at pH 45 but
`in both
`cases the maximum absolute value of WIkT
`is only of order 10 Not shown are local
`minima for pH 54 and 74 at s= 4 with
`WIkT = 0104
`The osmotic
`Eqs 6 to 11 is the sum of the contribu
`
`pressure calculated
`
`from
`
`tions from the excluded volume and from
`the region r > 2a + a in which the poten
`function is finite The effect of molecu
`tial
`lar size and shape
`on just
`the excluded
`volume contribution is shown in Fig 9 The
`lower three curves are for a sphere with
`molecular volumes of 10 to 20 x 105 A3
`molecule a range which circumscribes most
`
`two
`of the literature estimates The upper
`curves
`for a prolate ellipsoid with
`are
`vin fixed at 150 x 105 A3molecule and p
`= 32 or 4 Curve B which lies between the r
`data at pH 45 and 54 corresponds
`ellipsoid dimensions
`in Table II plus the
`repulsive contribution 012 15 A added to
`
`to the
`
`each dimension so that
`is effectively
`vin
`The
`increased to 150 x 105 A3molecule
`pH dependence
`the data could be ex
`of
`from
`plained in part by a shape
`change
`sphere to prolate ellipsoid with increas
`ing pH Such a shape change between pH
`45 and 54 has been suggested 43 but
`not substantiated by more recent
`investiga
`tions 37 38 44
`The contribution from the region r > 2a
`+ a was evaluated by numerical
`integration
`of Eqs 9 and 10 using fourth order Simp
`sons rule with the potential
`functions
`shown in Fig 8 To evaluate B3 the C2ri2
`Journal of Colloid and Interface Science Vol 79 No 2 February 1981
`
`is
`
`

`

`560
`
`VILKER COLTON AND SMITH
`and the inverse transform of C2k denoted
`by F1 can similarly be shown to be
`C2r = F1 e2k
`12k k sin krdk
`
`27
`
`27rr2 j9
`
`Numerical algorithms similar to those of
`Lado 49 were used to solve Eqs 26
`and 27 This solution for C2 was used in
`Eq 10
`The calculated estimates of B2 and B3 at
`each pH are tabulated in Table IV These
`results are relatively insensitive to t