Accepted Manuscript. Measurement of osmotic second virial coefficients by zonal size-exclusion chromatography. Donald J. Winzor

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1 Accepted Manuscript Measurement of osmotic second virial coefficients by zonal size-exclusion chromatography Donald J. Winzor PII: S (16) DOI: /j.ab Reference: YABIO 1354 To appear in: Analytical Biochemistry Received Date: 18 February 016 Revised Date: 30 March 016 Accepted Date: 4 April 016 Please cite this article as: D.J. Winzor, Measurement of osmotic second virial coefficients by zonal sizeexclusion chromatography, Analytical Biochemistry (016), doi: /j.ab This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 ABIO R1 Measurement of osmotic second virial coefficients by zonal size-exclusion chromatography Donald J. Winzor* School of Chemistry and Molecular Biosciences, University of Queensland, Brisbane, Queensland 407, Australia Running Title: Chromatographic measurement of virial coefficients *Fax: address: d.winzor@uq.edu.au

3 Abstract Numerical simulation of protein migration reflecting linear concentration dependence of the partition isotherm has been used to invalidate a published procedure for measuring osmotic second virial coefficients (B ) by zonal exclusion chromatography. Failure of the zonal procedure to emulate its frontal chromatographic counterpart reflects ambiguity about the solute concentration that should be used to replace the applied concentration in the rigorous quantitative expression for frontal migration: the recommended use of the peak concentration in the eluted zone is incorrect on theoretical grounds. Furthermore, the claim for its validation on empirical grounds has been traced to the use of inappropriate B magnitudes as the standards against which the experimentally derived values were being tested. Keywords: Thermodynamic nonideality Second virial coefficient Size-exclusion chromatography Bovine serum albumin Lysozyme

4 Introduction The determination of second virial coefficients by size-exclusion chromatography of protein solutions was reported and refined many years ago [1 3]. Those studies employed frontal chromatography [4.5] to allow unequivocal assignment of the protein concentration to which each measured elution volume referred. However, such attention to theoretical detail was deemed unnecessary in a much later investigation [6], which purportedly demonstrated the successful determination of second virial coefficients by zonal size-exclusion chromatography with the effective protein concentration taken as that corresponding to the maximum in the eluted zone. In view of the continual dilution that occurs during zonal migration [7,8], the validity of that claim certainly warrants further investigation. To that end, calculations based on an analytical solution for concentration-dependent migration with axial dispersion [9] reveal deficiencies in the zonal chromatography approach. Its limitations as an empirical procedure are also exposed by more critical assessment of the extent of agreement between experimentally determined and theoretically predicted second virial coefficients for the two proteins examined. This communication thus serves to re-emphasize the importance of employing frontal size-exclusion chromatography for the quantitative characterization of protein interactions a point stressed at the outset of such investigations [4,5]. Theoretical considerations The quantitative treatment of the effects of thermodynamic nonideality in size-exclusion chromatography was developed before general recognition of the need to consider the constraints pertaining to the definition of solute chemical potential under the experimental conditions being used [10,11]. It is therefore appropriate to consider this aspect first. 3

5 Definition of solute chemical potential in size-exclusion chromatography The equilibrium partitioning of solvent between the mobile and stationary phases of a sizeexclusion chromatography column ensures the identity of its chemical potential (µ 1 ) in the two phases. A similar situation applies to buffer and supporting electrolyte species, which may therefore be regarded as part of the solvent. Under those circumstances the chemical potential of the protein solute (µ ) is being monitored on the molar scale; and is described by the relationship [10,11] 0 0 (µ ) T,µ1 = ( µ ) T,µ1 + RT ln z = ( µ ) T,µ1 + RT ln (γ c /M ) (1) 0 where z is the molar thermodynamic activity and ( µ ) T,µ1 the standard state chemical potential of solute under the constraints of constant temperature and solvent chemical potential. In the second representation the thermodynamic activity of solute is replaced by the product of molar activity coeficient γ and the molar concentration, c /M,where c is expressed in g/l and M is the solute molar mass. Furthermore, a purely thermodynamic argument has established that the activity coefficient is described by the expression [10] γ = exp[b c /M +...] () where B is the osmotic second virial coefficient, a rigorously defined parameter that may be described in terms of physical interaction between pairs of protein molecules [1]. For a spherical solute with radius R the osmotic second virial coefficient can be expressed in terms of u (x), the potential-of-mean-force between two molecules separated by center-to-center distance x [10 1]; and hence calculated as 4

6 B = πn A (R ) 3 /3 f (x) x dx (3a) R in which f = exp[ u (x)/(kt)] 1 where k is the Boltzmann constant; and where Avogadro s number (N A ) is included to define the virial coefficient on a molar basis. For systems with a moderate ionic strength and relatively low net protein charge a reasonably accurate estimate of the second virial coefficient can be obtained from the expression [13] πN A R Z ( 1+ κr ) 1000 Z κ B = + + (4) 3 4I(1 + κr ) 18πN A I (1 + κr ) The first term is the hard-sphere contribution for a protein with Stokes radius R, and subsequent terms account for the additional excluded volume arising from charge charge repulsion between symmetrically distributed net charge Z : the factor of 1000 in the final term of Eq. (4) reflects the calculation of the Debye Hückel inverse screening length κ (in cm 1 ) as I from the ionic strength I measured on the molar scale. Quantitative treatment of nonideality in frontal size-exclusion chromatography Fortunately, the correct choice of concentration scale was made unwittingly in the original treatment of thermodynamic nonideality; and hence those expressions [1 3] retain validity [11]. (3b) Partition of the protein between mobile (α) and stationary (β) phases of a size-exclusion 5

7 chromatography column ensures the identity of its chemical potentials in the two phases, whereupon it follows from Eq. (1) that β α 0 α 0 β z / z = exp[{( µ ) ( µ ) }/( RT )] (5) On the other hand, the partition coefficient (σ) obtained experimentally from the elution volume (V e ) of a solute on a column (volume V tot ) with V o and V s as the respective mobile and stationary phase volumes via the expression σ = (V e V o )/V s = (V e V o )/(V tot V o ) = c β α (6) / c is the corresponding ratio of protein concentrations in the two phases. Inasmuch as the ratios of activities, z β α, and concentrations, c β α / z / c, are identical under thermodynamically ideal conditions, the experimental partition coefficient in the limit of zero protein concentration (σ 0 ) also defines the exponential of the difference in standard state chemical potentials [Eq. (5)]. Incorporation of Eq. () for the activity coefficient γ then gives [1,] σ = σ 0 α β exp[(b /M )(( c c ) = σ 0 exp[(b /M ) c α (1 σ)] (7) or, in logarithmic format [3,11] ln σ = ln σ o α + (B /M ) c (1 σ) (8) which allows determination of the second virial coefficient from the slope of a linear dependence of ln σ upon α c (1 σ). Further support for the use of Eqs. (7) and (8) has been provided by a subsequent demonstration of their validity for effects of thermodynamic nonideality reflecting nearest-neighbor interactions [14]. 6

8 Quantitative treatment of zonal migration in chromatography An obvious problem with the application of Eq. (8) to zonal size-exclusion chromatography data is the need for specification of the concentration to which V e (or its partition coefficient counterpart σ) refers. In the reported use of zonal size-exclusion chromatography [6] that concentration was calculated by combining the amount of protein applied with the width of the eluted zone at half-maximal height (concentration). For a symmetrical elution profile that approach merely provides an alternative estimate of the concentration at the peak of the zone (c p ). The validity of choosing this substitute for c has been examined by employing an analytical expression α for concentration-dependent zonal migration [9] to simulate the zonal chromatographic behavior of a solute for which partitioning is described by the isotherm σ = σ 0 + Kc By expressing the exponential term in Eq. (7) as a series expansion, it becomes σ = σ 0 [1 + (B /M ) c α (1 σ) +...] α 0 0 α = σ + (B / M ) σ (1 σ ) c [(B / M ) σ (1 σ )] ( c ) +... (9b) whereupon the consideration of partition in terms of linear concentration dependence suffices in situations where (B /M ) c α σ 0 (1 σ 0 ) << 1 to allow truncation of Eq. (9b) at the linear concentration term. Indeed, the same proviso (Kc << 1) is incorporated into the theoretical (9a) treatment of concentration-dependent zonal chromatographic migration. The analytical solution [9] refers to migration on a chromatography column with length L, through which solvent flows with linear velocity u. Zonal spreading is defined in terms of an origin 7

9 migrating with the linear velocity of a solute for which partition is governed by σ 0. In those terms that limiting solute velocity (U) and the extent of spreading from the median position (ξ) are given by the expressions [9] U = u/(1 + σ 0 /ε) (10a) ξ = l Ut (10b) in which ε is the column void fraction, V o /V tot, and ξ refers to the position in a zone at distance l from the redefined origin position Ut. Zonal spreading is quantified in terms of an axial dispersion parameter E c t, which for present purposes can be regarded as a fixed parameter whose magnitude can be assessed as that giving an appropriate estimate of the ratio of peak to initial concentrations (c p /c o ) for Ut = L; i.e., the value of c p /c o in the elution profile. The remaining fundamental parameter, λ, defined as λ = K/[ε(1 + σ 0 /ε)] = 4(B /M )σ 0 (1 σ) 0 / [ε(1 + σ 0 /ε)] (10c) incorporates the consequences of concentration-dependent partitioning on the length scale in which the other basic parameters are expressed. For a pulse solution input with length L o the analytical solution [9] may be written as where c exp(g) [erf(p + h) erf(q + h)] = c o 1 erf(p) + exp(g) [erf(p + h) erf(q + h)] + exp(m) [1 + erf(q)] (11a) p = (ξ + ½L o )/[ (E c t)] q = (ξ ½L o )/[ (E c t)] g = λc o Ut(ξ + λc o Ut + L o )/(4E c t) (11b) (11c) (11d) 8

10 h = λc o Ut/[ (E c t)] m = λc o UtL o /(E c t) (11e) (11f) The numerical simulation of normalized concentration distributions, c/c o as a function of ξ, thus entails the assignment of magnitudes to the length of the initial zone (L o ) with concentration c o as well as the column length (L = Ut), and the axial dispersion factor (E c t) for a specified value of λc o (product of the initial concentration and the parameter incorporating nonlinearity of the partition isotherm). Results and discussion In view of doubts already expressed about the validity of assigning a unique concentration to an elution volume obtained by zonal exclusion chromatography under conditions of concentrationdependent partition, this aspect is to be considered by employing Eqs. (11a f) to simulate zonal concentration distributions for a range of applied solute concentrations. However, it is first necessary to demonstrate the adequacy of such consideration of solute migration in terms of linear concentration-dependent partition. As noted in the previous section, a linear concentration dependence of protein partition requires the approximation that Eq. (9b) can be truncated at the linear term in c. The adequacy of this approximation is considered to be verified by results from α the initial characterization of nonideality by size-exclusion chromatography [1,]. Those results for ovalbumin on a column of CPG-75 beads equilibrated with phosphate chloride buffer (ph 7.4, I M) are presented in Fig. 1, together with the linear dependence (dashed line) predicted by the truncated Eq. (9b) with R =.9 and Z = 16 [1,] for this 44 kda protein. Although the experimental data are better described by the curvilinear dependence predicted by Eq. (7) [,15], the relatively small contribution from the higher-order terms in Eq. (9b) only comes into play at 9

11 concentrations greater than 0 g/l. In the present context the results for ovalbumin are considered to be described sufficiently well by linear concentration-dependent partition to warrant continuation with the simulation of zonal chromatographic migration on that basis. Simulation of zonal size-exclusion chromatography distributions Because Eqs. (11a f) are expressed in terms of normalized parameters, simulated distributions can be generated without reference to parameters for the particular protein on a given chromatographic column. The general features of zonal elution profiles reflecting linear concentration dependence of the partition isotherm are illustrated in Fig., where the patterns refer to distributions obtained for an applied sample length (L o ) of 0.1 cm on a 30-cm column (Ut = 30 cm) and values of 0.05 ( ), 0.04 ( ), 0.05 ( ) and 0.06 ( ) for λc o : E c t has been fixed at 0.14 cm in order to achieve an approximately 14-fold dilution at the peak of the distribution (c p /c o 0.07). Increasing the magnitude of λc o, which amounts to increasing c o for a particular protein on a given column, clearly leads to the progressive shift of the peak to the left; i.e., to progressive retardation of zonal migration. Such behavior is certainly the predicted qualitative consequence of partition governed by Eq. (9a). However, the possibility of retrieving the second virial coefficient (B ) by quantitative interpretation of the dependence of σ in terms of Eq. (7) or Eq. (8) [6] requires the consideration of a more specifically defined protein system: isoelectric lysozyme on the TSK G000SW column used by Bloustine et al. [6] has been used for that purpose. In that regard the respective values of 0.1 and 30 cm already assigned to L o and Ut reflect their application of 0 µl samples to a 30-cm TSK G000SW column with V o = 6.07 ml [6]. Consideration of the general zonal distributions (Fig. ) for a particular experimental system entails the interpretation of λc o in terms of a corresponding value of c o for that of λ defined by Eq. 10

12 (10c). To that end the results presented in Fig. 1 of Bloustine et al. [6] signify a limiting elution volume ( V ) of 9.85 ml for lysozyme on a TSK G000SW column with a mobile-phase (void) 0 e volume of 6.07 ml and a total volume (V tot ) of ml: σ 0 may therefore be taken as [from Eq. (6)], and ε = V o /V tot as The remaining parameter requiring specification is K = (B /M )σ 0 (1 σ 0 ) [see Eq. (10c)]. A value of 50 L/mol for B, or L/g for B /M, is obtained by substituting a Stokes radius (R ) of 1.7 nm for lysozyme [16] into Eqs. (3a,b) with Z = 0 for the isoelectric protein. The normalized concentration-dependence parameter (λ) for isoelectric lysozyme on that TSK G000SW column is thereby calculated to be L/g, whereupon c o = (λc o )/ g/l, and the applied concentration c app = c o /[ε + (1 ε)σ]. Appraisal of the zonal chromatographic procedure for virial coefficient determination Knowledge of the magnitude of c o for each distribution renders possible the conversion of the ordinate from a relative (c/c o ) to an absolute concentration scale (Fig.3A): the peak concentration c p ranges between 1.18 and.77 g/l in these simulated experiments with 15.7 c o 37.7 g/l. In order to check on the validity of the zonal procedure for determining second virial coefficients [6] the ξ value for each zonal peak in Fig. 3A has been used to define the length of column (L ξ) from which the zone peak would have eluted in the time commensurate with Ut = 30 cm, and hence to determine σ for each peak concentration c p. A plot of the resulting (c p, σ) data set according to Eq. (8) with c p substituted for c (Fig. 3B) reveals a 1.76-fold overestimate of B (88 α cf 50 L/mol) a finding that refutes the validity of the Bloustine et al. procedure [6]. The source of the problem with the approach is evident from Fig. 4, which depicts the progressive decline in c p /c o as the zone migrates down the 30-cm column a dependence that has been determined by employing Eqs. (11a f) to calculate c/c o distributions for the fixed initial zone length (L o = 0.1 cm) 11

13 and a range of column lengths (Ut values). Clearly, the greater extent of retardation that occurs during the early stages of migration cannot be neglected, this being the assumption inherent in the Bloustine et al. procedure [6]. A concentration greater than c p would need to be substituted for in Eq. (8). In summary, these simulation studies have failed to provide any support for the use of zonal size-exclusion chromatography to determine second virial coefficients. Instead they have demonstrated the invalidity of this recommended procedure [6]. There now remains the problem of rationalizing the experimental results that were purported to establish its validity. Reassessment of the claim for successful B determination The claim for validity of zonal size-exclusion chromatography as an empirical procedure for determining second virial coefficients relies upon purported agreement between estimates thereby obtained and values obtained from the literature. Inspection of Fig. 1 in the published report [6] certainly reveals a progressive increase in elution volume for lysozyme on a column of TSK G000SW in experiments with applied concentrations ranging between 14.9 and 65.3 g/l: the corresponding range for c p is g/l. Although a plot of those results according to Eq. (8) with c p substituted for c (Fig. 3 of [6]) is reasonably consistent with linear dependence of ln σ upon α c p (1 σ), the question at issue is the extent of agreement between the slope of that dependence and the theoretical value of B /M for lysozyme under the conditions of those experiments (ph 4.7, I 0.08 M). Because of the relatively high net charge (Z = 14) on lysozyme under these conditions α c [16,17], as well as the low ionic strength, the magnitude of B has been calculated as 500 L/mol (B /M = L/g) by means of Eqs. (3a,b). In that regard the same value can be inferred from 1

14 Fig. of Behlke and Ristau [16], which summarizes the ionic strength dependence of second virial coefficients obtained for lysozyme by sedimentation equilibrium. Failure of the predicted dependence of ln σ upon c p (1 σ), the solid line in Fig. 5A, to provide a reasonable description of the experimental data clearly refutes the claim for validity of the proposed procedure [6] as an empirical approach for the determination of second virial coefficients. A similar situation applies to the other system used to check the validity of the zonal chromatographic procedure for B determination bovine serum albumin on the same TSK G000SW column equilibrated with 50 mm phosphate buffer (ph 6., I 0.07 M). Indeed, the result for this system was probably considered to provide the main justification for use of the zonal procedure in that the slope of the dependence of ln σ upon c p (1 σ) essentially matched that obtained earlier [3] by frontal size-exclusion chromatography (Fig. 4 of [6]). However, the earlier result referred to isoelectric bovine serum albumin (B /M = L/g) rather than the negatively charged species (Z 0) being studied by Bloustine et al. [6]. Much poorer agreement is observed between theory and experiment (Fig. 5B) when the value of L/g for B /M that is calculated by means of Eqs. (3a,b) is used for the predicted dependence of ln σ upon c p (1 σ). Indeed, the observation of agreement between the estimates of B for bovine serum albumin by frontal and zonal size-exclusion chromatography is an indictment of the zonal technique: the slope of the plot of the zonal chromatographic data should have been greater because of the additional contribution to the second virial coefficient arising from repulsion between the negatively charged albumin molecules. Although the zonal size-exclusion chromatography experiments have sufficed to detect the concentration-dependent trends in elution volume reflecting the consequences of thermodynamic nonideality, their relative insensitivity to concentration-dependent migration [7] negates their use 13

15 for accurate quantification of the phenomenon. Indeed, the source of this quantitative imprecision is clearly evident in Fig. 1 of Bloustine et al. [6], where the maximum difference between V e and 0 V e is only 0.03 ml in elution volume measurements with a reported uncertainty of 0.01 ml [6]. The consequent uncertainty in experimental estimates of σ is incorporated into the abscissa as well as the ordinate parameter in Figs. 5A and 5B; and hence diminishes the prospect of accurate slope quantification. This imprecision in experimental delineation of the slope presumably accounts for the experimental underestimation of B by zonal size-exclusion chromatography inasmuch as overestimation was the predicted outcome of the numerical simulation studies (Fig. 3B). Concluding remarks Numerical simulation of zonal distributions reflecting linear concentration dependence of the partition isotherm has been used to invalidate a reported procedure for the measurement of osmotic second virial coefficients by zonal size-exclusion chromatography [6]. This finding is hardly unexpected in that the analysis merely entailed substitution of the zonal peak concentration for the applied concentration in quantitative expressions developed for rigorous determination of B by frontal size-exclusion chromatography [1 3]. In contrast with the situation prevailing in frontal chromatography, where the relevant concentration is unequivocally the applied α concentration c, the mobile-phase concentration of solute decreases progressively in zonal chromatography. Although frontal size-exclusion chromatography requires a relatively large amount of protein, a switch to the more economical zonal technique is not the solution to the problem of decreasing the amount of protein required for virial coefficient determination. Instead, cost savings in time and material need to be effected by decreasing the size of the column used for 14

16 frontal studies a modification now rendered feasible by technological developments in column chromatography. 15

17 References [1] L.W. Nichol, R.J. Siezen, D.J. Winzor, The study of multiple polymerization equilibria by glass bead exclusion chromatography with allowance for thermodynamic nonideality effects, Biophys. Chem. 9 (1978) [] R.J. Siezen, L.W. Nichol, D.J. Winzor, Exclusion chromatography of concentrated hemoglobin solutions: comparison of the self-association of the oxy and deoxy forms of the α β species, Biophys. Chem. 14 (1981) [3] K.E. Shearwin, D.J. Winzor, Thermodynamic nonideality in macromolecular solutions: evaluation of parameters for the prediction of covolume effects, Eur. J. Biochem. 190 (1990) [4] D.J. Winzor, H.A. Scheraga, Studies of chemically reacting systems on Sephadex. 1. Chromatographic demonstration of the Gilbert theory, Biochemistry (1963) [5] G.K. Ackers, T.E. Thompson, Determination of stoichiometry and equilibrium constants for reversibly associating systems by molecular sieve chromatography, Proc. Natl. Acad. Sci. USA 53 (1965) [6] J. Bloustine, V. Berejnov, S. Fraden, Measurements of protein protein interactions by sizeexclusion chromatography, Biophys. J. 85 (003) [7] D.J. Winzor, Reconciliation of zonal and frontal studies of concentration-dependent migration in gel filtration. Biochem. J. 101 (1966) 30C 31C. [8] E.E. Brumbaugh, G.K. Ackers, Molecular sieve studies of interacting protein systems. III. Measurement of solute partitioning by direct ultraviolet scanning of gel columns, J. Biol. Chem. 43 (1968)

18 [9] G. Houghton, Band shapes in nonlinear chromatography with axial dispersion, J. Phys. Chem. 67 (1963) [10] T.L. Hill, Theory of solutions. II. Osmotic pressure virial expansion and light scattering in two-component solutions, J. Chem. Phys. 30 (1959) [11] P.R. Wills, D.J. Scott, D.J. Winzor, The osmotic second virial coefficient for protein selfinteraction: use and misuse to describe thermodynamic nonideality, Anal. Biochem. 490 (015) [1] W.G. McMillan, J.E. Mayer, The statistical thermodynamics of multicomponent systems, J. Chem. Phys. 13 (1945) [13] D.J. Scott, P.R. Wills, D.J. Winzor, Allowance for the effect of protein charge in the characterization of nonideal solute self-association by sedimentation equiliibrium, Biophys. Chem. 149 (010) [14] D.J. Winzor, P.R. Wills, Allowance for thermodynamic nonideality in the characterization of protein self-association by frontal exclusion chromatography: hemoglobin revisited, Biophys. Chem. 104 (003) [15] D.J. Winzor, Analytical exclusion chromatography, J. Biochem. Biophys. Methods 56 (003) [16] J. Behlke, O. Ristau, Analysis of the thermodynamic nonideality of proteins by sedimentation equilibrium experiments, Biophys. Chem. 76 (1999) [17] P.R. Wills, D.R. Hall, D.J. Winzor, Interpretation of thermodynamic nonideality in sedimentation equilibrium experiments on proteins, Biophys. Chem. 84 (000)

19 LEGENDS TO FIGURES Fig. 1. Concentration dependence of partition coefficients obtained [1] by frontal size-exclusion chromatography of ovalbumin on a column of CPG-75 equilibrated with phosphate chloride buffer (ph 7.4, I 0.156), together with their predicted description ( ) in terms of the linear dependence predicted by Eq. (9b) truncated at the linear term in c α. (Data taken from Fig. 1 of [].) Fig.. Normalized concentration (c/c o ) distributions [calculated via Eqs. (11a f)] after migration of a 0.1-cm initial zone for 30 cm] showing the progressive retardation with increasing values of 0.05 ( ), 0.04 ( ), 0.05 ( ) and 0.06 ( ) for the concentration-dependent partition parameter λc o : the abscissa, ξ = l Ut, is defined in terms of distance from the exit plane of the column. Fig. 3. Simulated size-exclusion chromatographic behavior of a 0.1 cm initial zone of isoelectric lysozyme on a 30-cm column of TSK G000SW: σ 0 = 0.804, B /M = L/g. (A) Concentration distributions derived from Fig. on the basis that c o = (λc o )/ g/l (see text): numbers adjacent to each distribution indicate the initial concentration (g/l). (B) Plot of the (c o, σ) data set according to Eq. (8) with c p substituted for c : the broken line is the best-fit linear description, whereas the solid line is the dependence predicted on the basis of the theoretical value for B /M. Fig. 4. Calculated variation, via Eqs. (11a f), of the zonal peak concentration (relative to its initial α value) with distance migrated down the 30-cm column: L o = 0.1 cm, λc o = 0.04 (see Fig. 3). Fig. 5. Application of Eq. (8) with c p substituted for α c to determine second virial coefficients by zonal exclusion chromatography. (A) Experimental data [6] for lysozyme on a 30-cm column of 18

20 TSK G000SW equilibrated with 50 mm acetate buffer (ph 4.7, I 0.08 M), together with the predicted dependence (slope B /M ) for lysozyme under these conditions. (Experimental data taken from Fig. 3 of [6].) (B) Corresponding comparison for bovine serum albumin in 50 mm phosphate buffer (ph 6., I 0.07 M). (Experimental data taken from Fig. 4 of [6].) 19

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