Biotechnol. Prog. 1991, 7, 544-553
544
Nonlinear Gradient Isotherm Parameter Estimation for Proteins with Consideration of Salt Competition and Multiple Forms Roger D. Whitley,?James A. Berninger, Nathalie Rouhana, and N.-H. Linda Wang* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
Salt gradients in ion-exchange chromatography are routinely used to speed separation of proteins and to concentrate products, but systematic optimization of these gradients requires protein equilibrium data as a function of salt concentration. An understanding of conformational changes, aggregation, and salt effects, which include both competition and affinity modulation, is important for equilibrium isotherm parameter estimation. In this study, gradient elution of bovine serum albumin (BSA) in anion exchange was well predicted by a salt-modulated nonlinear isotherm which considers salt competition. The isotherm was able to predict BSA gradient elution from batch equilibrium data. The same isotherm was also able to predict elution for various gradient slopes when fitted to an intermediate slope gradient experiment. If multiple forms due to aggregation or denaturation exist, isotherm parameters are readily averaged in batch experiments because of the long equilibration times. Similarly, gradient experiments yield averaged parameters because the salt gradient tends to merge the closely eluting forms. However, in isocractic elution, if the reaction rate is not rapid enough to give a merged peak, the estimated isotherm parameters are only fair predictors of gradient behavior and vice versa. Slower flow rates in isocratic elution can help reduce the discrepancy by allowing forms to merge through interconversion. As an alternative to determining averaged parameters, consideration of two binding forms, using VERSE-LC,an advanced rate model, gave good agreement with experimental data over the entire range of salt gradient durations.
1. Introduction Purification of proteins is often accomplished by salt gradient elution on an ion-exchange column. The salt gradients speed separation and concentrate products, but systematic optimization of these gradientsrequires protein equilibrium data as a function of salt concentration. To date, some researchers have tried to predict gradient elution behavior of proteins on the basis of a series of isocratic experiments, but even the use of empirical correction factors has been of limited success because secondary effects of salt competition and salt-induced conformational changes of proteins have not been considered in the parameter estimation and modeling of these systems. 1.1. Prior Studies Related to Gradient Chromatography. In 1952, Alm et al. introduced gradient elution analysis as a tool to reduce band spreading associated with substances having Langmuirian isotherms. Experimental demonstration was on chromatography of oligosaccharides (Alm, 1952). Today, gradient elution techniques are widely used in analytical- and preparative-scale separation of proteins and other biochemicals (Lee et al., 1978; Yamamoto et al., 198313; Gooding and Schmuck, 1985). Many researchers have attempted to explain the behavior of gradient data through various models. The developments leading up to current understanding of salt gradient elution of proteins and the shortcomings of current models are discussed here and summarized in Table I. If a system operates only in the linear isotherm region and if there are no reactions present, theories exist to guide one in the design of separations (Stout et al., 1986; Ghrist
* To whom correspondence should be addressed. +
Current address: Air Products and Chemicals, Inc., Allentown,
PA 18195-1501. 87567938/9 1/3007-0544$02.50/0
et al., 1988; Ghrist and Snyder, 1988a,b; Kang and McCoy, 1989). Snyder and co-workers (Snyder, 1980; Snyder et al., 1983,1988; Eble et al., 1987; Cox et al., 1988) have put forth a model for multicomponent systems which assumed a linear relation between the log of the capacity factor and the solvent strength (LSS). Snyder measured deviations in peak symmetry in gradient and in isocratic elution to determine generalized correlations for changes in band shape. These correlations, along with the assumption of an LSS gradient, are only valid for small solutes within limited operating conditions, and discrepancies occur at high column loadings (Czok and Guiochon, 1991). Experimental verification of retention time correlations has been performed for proteins in multicomponent reversed-phase systems but limited to low loading conditions (Stout et al., 1986; Cox et al., 1988). Schoenmakerset al. (1978) have also provided local equilibrium equations for the determination of retention time of low molecular weight solutes in gradient elution for various functional relationships between the capacity factor and the volume fraction of the solvent. There is good agreement between the gradient-estimated parameters and the isocratic experimental results, although small systematic differences exist. However, the systems studied had linear isotherms and do not exhibit complex properties often associated with protein systems, such as extreme tailing and multiple peaks for each solute. More detailed mathematical models of gradient elution have been studied by Yamamoto et al. (1983a,b) for ionexchange systems undergoing a salt gradient and by Antia and Horviith (1989) and El Fallah and Guiochon (1991) for reversed-phase systems. Yamamoto et al. (1983a,b, 1987,1988)have proposed a plate theory for linear isotherm systems to describe salt gradients in ion-exchange chromatography where the change in ionic strength in each
0 1991 American Chemical Society and American Institute of Chemical Engineers
Bbtechnol. hog.., 1991, Vol. 7, No. 6
545
Table I. Toward a More Detailed Understanding of Gradient Elution of Proteins in Ion Exchange ~~
topic of research introduction of gradient elution to chromatography analysis of reversed-phase gradient elution linear solvent strength (LSS) model stage model for ion-exchange gradient elution mechanism of protein folding gradient elution for preparative systems experimental studies of gradient ion exchange calculation of molecular weights under gradient elution
results pertinent to salt gradient elution improvement in resolution and reduction of band spreading for Langmuir-type isotherms experimental demonstration with oligosaccharides development and experimental verification of capacity factor to mobile-phase relationships for small alcohols development of simple, empirical relationships between solute elution and solvent characteristics experimental verification for proteins required empirical correction factors agreement for proteins at linear isotherm concentrations, low mass transfer resistance systems linear relationship between logarithm of distribution coefficient and logarithm of ionic strength evidence of intermediate forms in the protein folding process unfolding process often accompanied by the formation of aggregates modulator dependence of nonlinear isotherms and separation factors investigated for multicomponent systems protein gradient parameters varied with gradient duration due to conformational changes LSS model unable to accurately account for effects of conformational changes on sorption dynamics BSA molecular weight varied as a function of gradient slope (salt-dependent aggregation)
plate is considered. Mass transfer effects are represented solely by the number of plates, which is held constant. The work by El Fallah and Guiochon (1991) centered on a small solute (2-phenylethanol) and used a Craig model which considers nonlinear isotherm effects in multicomponent systems. In the analysis by Antia and Horvdth (1989), a differential mass balance model with a lumped dispersion coefficient is used. Both the number of plates and the dispersion coefficient can change with particle size, flow velocity, diffusivity, and, in nonlinear systems, with solute concentration and pulse size. Even if no chromatographic information is available, there is a computer-based simulation package which can assist in the design of multicomponent gradient separations for peptides of known amino acid sequence up to 40-50 units (Hodges et al., 1988). This simple model is useful for small to medium peptides since the three-dimensional structure of the resulting peptide is usually not complex enough to give unusual retention behavior. The limitations of this type of model are that linear isotherms are used (competition among solutes is not considered) and reactions and mass transfer effects are not considered. Salt gradients affect solute retention in two basic ways. The salt is (usually) a weak competitor for exchanger sites. An increase in salt concentration may force the solute out more quickly because of increased competition for sites. Another effect, somewhat more subtle in action but often more noticeable in effect, may be a salt-induced change in protein conformation. As a result, the protein may bind with an altered structure and affinity (Hearn and Anspach, 1990). Hearn et al. (1988) noted that the LSS theory becomes invalid if the physiochemical basis of isocratic and gradient elution is not the same. Hodder et al. (1989, 1990)have studied the retention and bandwidth properties of proteins as a function of various displacer salts under gradient conditions. By comparison of retention plot slopes for various salts, they concluded that different salts seem to cause binding by different sites on the protein. Hodder et al. also found that isocratic and gradient retention data were not superimposable. Estimated gradient parameters varied with gradient duration, indicating that physiochemical changes, such as conformational changes or aggregation, were being affected by the gradients (Hodder et al., 1989). Goldberg (1985) presented evidence that intermediate forms exist in the protein folding process and that the unfolding process is often accompanied by the formation
references Alm et al. (1952), Alm (1952) Schoenmakers et al. (1978) Snyder (19801, Cox et al. (1988), Snyder et al. (1989) Stout et al. (1986) Yamamoto et al. (1983a,b)
Goldberg (1985) Antia and Horvlth (1989) Hearn et al. (1988), Hodder et al. (1989,1990) Mhatre et al. (1990)
of aggregates. Gooding and Schmuck (1985) noted the appearance of multiple peaks for gradient elution of bovine serum albumin (BSA)under salt gradients in strong anionexchange HPLC. These peaks were attributed to aggregation of BSA. Mhatre et al. (1990) found that the molecular weight of BSA varied as a function of gradient slope, adding further evidence of aggregation. The effects of protein aggregation on isocratic elution have been explored previously (Whitley et al., 1991; Van Cott et al., 1991). 1.2. Objectives. The overall goals of this research are to quantify the binding affinity of protein systems undergoing salt gradients and to study the competition among the various binding forms and the salt. The question of whether or not isocratic isotherms can be used to predict gradient behavior and vice versa will be investigated. From an understanding of these interactions, affinity dependencies, and the statistical behavior of the models, estimability of isotherm and mass transfer parameters will be discussed. The presence of multiple forms of BSA and of conformational changes in BSA under gradient conditions will be investigated, focusing on the complications in isotherm parameter estimation. 1.3. Scope and Method. In this study, a stage model incorporating mass transfer effects and an equilibrium isotherm (Whitley et al., 1989b) is used in the analysis of protein gradient elution data in ion-exchange chromatography. Isocratic and salt gradient elution experiments were performed on BSA solutions to obtain equilibrium isotherm information. Some methods for determining isocratic nonlinear isotherm parameters have been reported (Cowan et al., 1989; Whitley et al., 1989a,b). The work in this paper builds upon our statistical analysis of nonlinear isocratic systems in order to establish a framework for estimating nonlinear isotherm parameters from gradient elution data. A detailed rate model is used to study multiple protein forms and their effects on isotherm parameter estimation. A modified multicomponent Langmuir equation incorporating modulator (such as a salt) concentration was used (Antia and Horvdth, 1989). The Langmuir equation was further modified in this work to consider salt competition. Experiments and simulations were performed to illustrate the behavior of a binary system undergoing gradient elution for a wide range of gradient durations. 1.4. Major Results and Conclusions. The use of a salt-modulated isotherm equation, including salt compe-
Biotechnol. Prog., 1991, Vol. 7, No. 6
540
tition, provided a good description of isotherm data over a wide range of salt concentrations. A series of isocratic pulse elution experiments cannot, however, accurately predict both elution time and peak sharpness for salt gradient elution of proteins if aggregation or conformational changes are not considered. Sensitivity analysis showed that gradient isotherm parameters could be accurately estimated if experiments were carried out in the nonlinear portion of the isotherm. Isocratic elution at low salt concentration and long gradient elution showed a second peak, indicative of a nonequilibrium process such as a conformational change or aggregation. Application of a single-form model to such data resulted in errors in parameter estimation. Allowing for salt competition improved model agreement with the isocratic elution data. Consideration of multiple protein forms by an advanced rate model allowed one set of isotherm parameters to accurately predict isocratic and gradient elution behaviors for a wide range of gradient durations. Therefore, an understanding of secondary effects, such as conformational changes or aggregation, in gradient elution is vital for accurate isotherm parameter estimation and for accurate scale-up.
2. Theory 2.1. Stage Model Development. The basic development of the stage model used in this work was presented by Whitley et al. (1989b). The major additions to this work are the introduction of gradients and multicomponent isotherms so that the effects of modulator concentration change under gradient elution can be explicitly represented. This section begins with a brief summary of the assumptions and features of the stage model. The stage model treats the column as a series of Nu equal volume (V) stages. The mobile phase flows through these stages at a rate F. Within each stage the total volume is divided into three fractions: the bulk solution (VEb), the pore solution [ V(1 - tb)Ep], and the solid phase [ V(1 - € b ) ( l - e,)]. For a given stage, a gradient exists between the bulk solute concentration, cb, and the pore solute concentration, C,. This difference results from the combined effects of the film and pore diffusion mass transfer resistances. The solute concentration in the pores is taken as an average value. Equilibrium, assumed between the solute in the pore solution and the solute on the solid phase, is described by either the Langmuir isotherm or the counterion modulator (CIM) isotherm (Antia and Horvlth, 1989). The Langmuir isotherm (eq 1)has been applied to biological systems
cpi=
ponent Langmuir and the physical origins of the isotherm interactions. In a subsequent paper, Guiochon's group (Lin et al., 1989)proposed a modified Langmuir isotherm which incorporates interaction terms. Unfortunately, this modification requires seven parameters to model a pair of solutes, where the original multicomponent form only requires four parameters. The usefulness of the modified form is uncertain for the systems under study. For salt gradients in ion-exchangechromatography, Antia and Horvlth (1989) stated that the CIM isotherm (eq 2)
a 0 i4-zoicp,
cpi= +
N,
(2)
1 xboj$-z'jCpj j=1
gives good agreement with experimental gradient data and it is used as a starting point for gradient modeling in the current study. Additionally, a modified CIM isotherm is proposed to include salt competition (see eq 14 under Results and Discussion). In chromatography three types of mass transfer effects contribute to band spreading of a pulse: axial dispersion, film mass transfer, and pore diffusion. Axial dispersion results from path tortuosity and from wall effects (flow becomes slow very close to surfaces). The effects of axial dispersion can be well described by adjusting the number of stages used in this model. For the current study, however, 250 stages were used, implying that axial dispersion is negligible for this system (Whitley et al., 1989b). Transport of solute from mobile phase to particle surface and subsequent diffusion in the pores can be represented by film mass transfer and pore diffusion terms. If the concentration profile in the resin pellet is parabolic, then the two phenomena may be represented by a simple linear driving force consisting of the difference between the bulk and pore concentrations weighted by an overall mass transfer coefficient (Cen and Yang, 1986). The main limitation of the stage model is that this mass transfer coefficient, kL, can change with the pulse concentration, pulse size, column length, flow rate, and particle size (Whitley et al., 1989b). Even with these dependencies, estimates of isotherm parameters by the model were still shown to agree well with experimental data (Whitley et al., 1989b). For the purpose of estimating isotherm parameters, this lumped parameter approach has the advantage of simplicity and no need for several mass transfer parameters. On the basis of the above assumptions, mobile phase and resin phase mass balances for stage n give
aiCpi N"
for several decades (Scatchard et al., 1950). The form of the equation expresses a simple competition between various species in which the bound concentrations are affected by the change in the pore concentration of all the species over the exchanger sites. Exchanger capacity is implicit in the a and b parameters. This isotherm, written for just one component, is particularly well suited for describing amino acid and protein binding affinity under isocratic conditions (Whitley et al., 1989a,b) and will be used to model isocratic experiments in the current study. However, this isotherm cannot represent the interaction of competitive isotherms for certain protein systems. One such example is presented by Huang and Guiochon (1989), in which they explore the limitations of the multicom-
(4) Substituting
into eqs 3 and 4, one obtains
547
Blotechnd. Rog., 1991, Vol. 7, No. 0 0.03
0.025 0.02 0.015 0.01
0.005 0
-)
dC,n d C , n d9,,n dt +
dm,x]/[
‘p(’
-‘b)
0
25
0
5
50
la,
75
125
150
175
200
+
For a rectangular pulse input, duration dp, the initial and boundary conditions are
Subjected to the initial and boundary conditions given by eq 10, eqs 8and9 are solved numerically by “a differential/ algebraic system solver” (DASSL) to determine the response (Petzold, 1982). Calculationsare performed with 32 bits of precision, and all numerical tolerances are set to ensure six significant digits in model output. For any given set of isotherm and mass transfer parameters, the model predicts a unique effluent history. Parameter values are estimated by minimizing the error between the experimental and the model response curves using the IMSL (International Mathematical and Statistical Libraries, Edition 10, Houston, TX) routine DUNLSF, which employs a finite difference Levenberg-Marquardt algorithm with strict descent. The criterion for minimization is defined as
where m is the number (between20 and 50) of experimental points used from the effluent history curve of the protein. A fit based on absolute error works better than one based on relative error since a relative error fit gives too much emphasis to the tail of the pulse. Some of the tailing may be caused by extra column effects and slow kinetics at the surface. These effeds are effectively lumped into kL. 2.2. VERSE-LC Model. The experimental results reported below showed evidence of multiple protein forms as a result of conformational changes or reactions. To study this phenomenon in greater depth, a more detailed model was needed. To this end, VERSE-LC (VErsatile Reaction-SEparation for Liquid Chromatography) has been used for those simulations. VERSE-LC was developed to consider detailed mass transfer, gradient elution, solution- and solid-phase reactions, and slow sorption kinetics (Whitley, 1990; Berninger et al., 1991). Mass transfer (Lee et al., 1989) and reaction (Van Cott et al., 1991; Whitley et al., 1991) aspects of the model have been verified previously. Gradient capabilities of the model will be verified with the current experimental data.
3. Experimental Procedures BSA (67 OOO g mol-l; p1 = 5.071, crystallized and essentially globulin-free,was purchased from the Sigma Chemical Co., St. Louis, MO. Q-Sepharose Fast-Flow exchanger (quaternary amine functional group, strong anion exchanger) was donated by Pharmacia Fine Chemicals of Uppsala, Sweden, and consists of an agarosematrix cross-linked with 2,3-dibromopropanol. For the BSA and Q-Sepharose system, q, = 0.35 and ep = 0.57 (Brown,1990). The macroporous particles are spherical with a diameter range of 45-165 pm (Pharmacia, 1989). These exchangers display negligible bed volume change upon changes in ionic strength.
0.3 0.25 0.2 0.15 0.1 0.05 0 10
15
‘20
25
30
35
e
Figure 1. Isocratic elution of BSA on Q-Sepharose at a p H of 8.00 for salt concentrations of 0.157 (A), 0.207 (B),and 0.248 (C) N and model predictions using (a) direct isocratic fit, (b) CIM gradient average estimates, (c) CIM with salt competition (“combined”data), (d)CIMtwo-formestimate,and (e) CIMtwoform estimate with interconversion. (a-)
Trizma hydrochloride (Trizma-HC1)was the buffer for BSA solutions at pH 8.0. The column apparatus was a gift from Pharmacia of Uppsala, Sweden. The inner column diameter is 1cm, and the column is equipped with plungers on both ends so that length may be varied from 0 to 17 cm. A l-mL injection loop was used for all experiments. Rheodyne sample loops and a three-way valve were purchased from Alltech Associates, Inc., Deerfield, IL. A Spectroflow430 low-pressuregradient former (KratosAnalytical, Ramsey, NJ) provided linear salt (Clcounterion) gradients by mixing two reservoirs uniformly. The resulting solution was pumped by an ISCO WIZ peristaltic low-pressure pump (Lincoln, NE). Detection of the solutes was by flow through a quartz flow cell (l-cm path length) which was placed in a Lambda 3A UV/vis spectrophotometer (Perkin-Elmer, Oak Brook, IL). The column packing, isocratic elution, and gradient elution procedures are routine, and discussion is provided by Rouhana (1990).
4. Results and Discussion This section will be subdivided into four parts: (1) isocratic elution at three different salt concentrations, including the fitting of the isocratic experiments and attempts to predict gradient elution from the isocratic experiments; (2) gradient elution, includingdirect isotherm fitting of the gradient experiments, attempts to predict isocraticelution from gradient experiments, and sensitivity analysisof the CIM isothermparameters; (3) consideration of salt competition; and (4) a discussion of fundamental differences between gradient and isocratic elution when a protein exists in multiple forms due to conformational changes or aggregation. 4.1. IsocraticElution. 4.1.1. DirectFitofIsocratic Experiments. Three isocratic elution experiments were conducted for BSA at salt levels of 0.157,0.207, and 0.248 N (Figure 1);specific experimental conditions are shown in Table 11. The procedure for estimating simple Langmuir isotherm parameters from isocratic elution experiments has been previously developed and verified (Whitley
548
Biotechnol. Prog., 1991, Vol. 7, No. 6
Table 11. Experimental Conditions for Isocratic and Gradient Elution Cdt, N
0.157 0.207 0.248 gradients (0.157 to 0.243)
L,cm 3.81 4.05 3.80 4.35
F,mLmin-I 1.35 1.40 1.40 1.38
C,,M 1.488 X 10" 1.513 X 10"' 1.449 X lo4 1.488 X lo4
Table 111. Determination of Isotherm and Mass Transfer Parameters from Isocratic Pulse Data Lt,N a b, M-1 a / b, M k ~ = * ,cm min-' 0.157 31.60 1.166 X lo5 2.710 X 10" 0.01631 4.070 2.528 X 10' 1.610 X 10" 0.207 0.01878 0.248 1.506 1.977 X lo4 7.618 X 0.00993
0
5
10
15
20
25
e
Figure 3. Attempts to fit CIM isotherm to l-min gradient data (-) using (a) combined estimates, (b) direct fit to gradient, (c) gradient average estimates, (d) two-form estimate, and (e) salt effluent history for two-form estimate.
0.15j 0.1
0.1
0.15
0.2
y \\ ~,
,
,
25
30
0
Ga~cIN1
5
10
20
15
oh
0.2
35
c
r
0.1
p 0.15
0.25
0.2 cuii INI 0.19 0.18 0.17 0.16 0.15
0.25
C*d,[Nl
Figure 2. Comparison of Langmuir a and b estimates for BSA,
pH 8, from various sources: batch equilibrium method, Whitley e t al., 1989a ( 0 ) ;isocratic pulse elution method, Brown, 1990 (+); current elution study (A). The dotted line is CIM best fit and the solid line is best fit for modified CIM with salt competition.
et al., 1989b). Using that procedure, we determined the parameters shown in Table 111for BSA at the three salt levels, and the associated model predictions are shown as solid lines (curves a) in Figure 1. The resulting capacities (Table 111)are 25-50 5% of those determined by batch equilibrium estimation on this system (Whitley et al., 1989a). The lower capacity is probably due to the presence of a denatured form or aggregates, the effects of which would be averaged in batch parameter estimates. Figure 2 shows a comparison of fitted Langmuir isotherm parameters from three sources, including both batch equilibration and isocratic elution methods. There is a smooth decrease in both a and b with salt over the fairly wide salt range covered. Agreement is good for the two different methods and among the three experimentalists. 4.1.2. Estimation o f Gradient Isotherm Parameters from Isocratic Experiments. A series of gradient elution experiments for gradient times ranging from 1to 25 min are shown in Figures 3-7 (experimental conditions shown in Table 11). Increasing the gradient slope shows the expected trends-the BSA peak elutes earlier and is sharpened. The BSA peaks are somewhat asymmetrical, even under the sharpening effect of the gradient. The small hump eluting at approximately 6 min is not affected by gradient slope; it is considered an impurity and will not be used in fitting the gradient elution curves. By comparison of the CIM isotherm (eq 2) to the simple Langmuir (eq l),one can derive the following relation-
6
4
io
i
i5
2'5
io
0.24
i5
9
Figure 5. Attempts to fit CIM isotherm to 10-min gradient data
(-.) using (a) combined estimates, (b) direct fit to gradient, (c) gradient average estimates, (d) two-form estimate, and (e) salt effluent history for two-form estimate.
ships:
Thus, from the series of a and b as functions of salt in Table 111, one can calculate the best fit values of aoi, boi, and Zoi for eqs 12 and 13. The Eureka numerical solver package, version 1.0 (Borland Software, Scotts Valley, CA, 1987)was used for this fitting, and the resulting values are shown in Table IV as "isocratic". This procedure is a first approximation for determining the salt gradient parameters. As a comparison, CIM parameters for all the data from Figure 2 were also fitted to eqs 12 and 13(rowdenoted as "combined" in Table IV). The combined fitting of Figure 2 data gives good agreement to the wide range of experimental data (ZO,Figure 2). The large differences between parameter estimates for the higher salt range (isocratic) and the full salt range (combined) are due to omission of salt competition in the CIM isotherm equation and possibly multiple protein binding forms arising from conformational changes or aggregation. As a result, neither set of parameters gives good agreement with all three isocratic curves of Figure 1 (curves not shown).
549
Biotechnol. Prog., 1991, Vol. 7, No. 6 0.15
0.25 0.24 0.23 0.22 0.21 0.2 Cmi,[Nl 0.19 0.18 0.17 0.16 0.15
C
0.1 C
0.05
0 0
5
10
20
15
25
30
45
40
35
e
Figure 6. Attempts to fit CIM isothermto 15-mingradient data using (a) combined estimates, (b) direct fit to gradient, (c) gradient average estimates, (d) two-form estimate, and (e) salt effluent history for two-form estimate. (-a)
0.1
,
,
0.25
-0.18
o
5
io
I
I
I
I
zo
I
I
,
15
25
30
35
I
...
I
,I
40
45
I
-0.17 -0.16
.I
0.15-
50
e
Figure 7. Attempts to fit CIM isotherm 25-min gradient data using (a) combined estimates, (b) direct fit to gradient, (c) gradient average estimates, (d) two-form estimate, and (e) salt effluent history for two-form estimate. (-e)
Table IV. Comparison of Estimation Methods for Gradient Parameters
isocraticD combined* 1-min grad. 5-min grad. 10-min grad. 15-min grad. 25-min grad. grad. avg
1.25 X 1.11 X 6.24 X 6.23 X 6.22 X 5.96 X 6.44 X 6.22 X
loW4
1.20 37.46
3.58
4.73
4.74 4.00
3.50
6.460 3.930 5.755
5.770 5.764 5.770 5.753 5.762
0.0150 0.0309
0.0189
0.0139
0.0160
1.10 1.10 2.80 1.10
0.0144
4.11 0.0194 1~52~ Fit from eqs 12 and 13 for all three salt levels. The gradient average value of k h t was used in the simulations, since it was not estimated from isocratic data. Fit from eqs 12 and 13 for all the data of Figure 2, including both batch and pulse data. Not including the 25-min gradient estimate for k h .
If one desires to average out the scatter in parameter estimates arising from these phenomena, all the data of Figure 2 (isocratic and batch) can be used to predict the series of gradient elution curves. Comparisons of these isocratic predictions to the actual gradient data are shown in Figures 3-7 (curves a). At first glance, one can see that even the combined estimates of CIM paramaters give a poor prediction of gradient behavior. The predicted curves elute too late and do not show increased focusing for increasing gradient steepness. Thus, the simple CIM isotherm cannot model the BSA system as a single protein form over a wide range of salt concentrations or under gradient conditions. 4.2. Gradient Elution. The gradient parameters can be mathematically determined from a series of isocratic runs, as was shown in the previous section. The gradient, however, is not a series of isocratic runs but rather is a continuous increase of salt concentration during the experiment. Consequently, the elution curve shape is a result of this continuous increase in salt concentration. It will be shown in this section that a direct fit of the CIM isotherm to gradient elution data gives a better prediction of gradient elution behavior than does a series of isocratic elution experiments.
4.2.1. Direct Fit of Gradient Experiments. The direct-fit CIM isotherm parameters for several gradient times are shown in Table IV. Figures 3-7 show the direct fita for various gradient times (curvesb). For each gradient time, direct fitting of the gradient elution gives a much better fit of the experimental curve than the combined prediction. Both elution time and curve shape are well estimated. The most striking results shown in Table IV are that by doing one linear gradient pulse experiment, one can get very good estimates of uoi and Zoi and that these values are unaffected by variations in the lumped mass transfer parameters. The continuous change in salt concentration makes the CIM term +-zoc nonconstant, which improvesthe estimability of Zoi over that from fitting a series of isocratic experiments. The value of boi differs quite a bit between gradient and combined estimates but relatively little between various gradient slopes. Peak merging under gradient tends to result in an averaged value of boi. 4.2.2. Comparison o f Gradient Fits for Various Gradient Slopes. Looking a t Figures 3-7 again, one can see the gradient elution predictions for one gradient time using the averaged parameters from all gradient estimates (curvesc). Table IV indicates that direct fitting of different gradient durations will give different estimates for the isotherm parameters. The boi parameter does show some dependence on the gradient duration, a consequence of salt competition and conformational changes. As noted earlier, the BSA system has extreme tailing and shouldering for 0.157 N isocratic elution (Figure 1A) and for longer gradient elution cases (Figure 7), which indicate the presence of two or more forms. Treating this system as having one form (as a first attempt at parameter estimation) makes the fitted parameters dependent on gradient slope. In minimizing the error between experimental data and model, the isotherm values represent an average of the values for the forms actually present. 4.2.3. Prediction of Isocratic Parameters from Gradient Isotherms. Since the simple Langmuir isotherm represents a degenerate case of the CIM isotherm, which explicitly considers salt modulation, it may be possible to use gradient experiments to predict isocratic elution. Looking at Figure 1 (curves b), one can see the CIM predictions for the three isocratic elution experimenta (using the average gradient parameters shown in Table IV). The CIM gives a poor prediction of isocratic behavior and tends to predict earlier elution than the data for low salt and later elution than the data for high salt, partly because salt competition is not considered and partly because only one form is considered. 4.2.4. Sensitivity Analysis of Gradient Isotherm Parameters by Contour Plots. A general discussion of contour analysis is presented by Whitley et al. (1989b). Contour plots are useful for showing the sensitivity of a model for various parameters, which correlates with the ability of those parameters to be estimated from experimental data. Simulation parameters for the contour plots are a rough, rounded average of the gradient parameter boi = 4.0 M-l, Zoi estimates of Table I11 ( u O ~ = 6.2 X = 5.8, hdt = 1.0 cm min-', and hssA = 0.015 cm m i d ) . Some of the coarseness of these contour plots arises from the relatively low number of simulations used to produce them. There are 11parameter values on each axis, for a total of 121 simulations per contour plot. Since a nonlinear isotherm is being used in the model, the first question to be answered is whether column concentrations of BSA are actually in the nonlinear portion of the isotherm. The boi versus uoi contour plot shown in Figure 8 for the 10-min gradient shows a trough at about 6 5 O from the uoi axis, with a minimum near the center of
Biotechnol. Prog., 1991, Vol. 7, No. 6
550 I
10
I
-
I
I
\Y\\ \ I
I
zv
I
i-
b
Table V. Parameter Estimates for CIM with Competition source a* b0,M-' isocratic elution data, 1.177 X lo4 60.66 Figure 1 combined isocratic and 2.530 X 10" 164.1 batch equilibrium data, Figure 2 5-min gradient elution 5.993 X lo4 3.874 data, Figure 4 0.35 0.3JA
1\1
i
b
Salt
zo,
zob
6.722
4.020
4.757
3.104
5.696
5.551
h
io
i5
io
is
io
35
40
45
io
15
20
25
30
35
40
45
50
0.35 0.3 0.25
-10
0
10
Xd
Figure 8. Contour plot of bo vs ao for the CIM isotherm and a 10-min gradient.
the plot. This is analogous to Figure 9d in Whitley et al. (1989b) and indicates nonlinear concentrations and good ability to estimate aoi and b'i. For a gradient duration of 1 min, the trough is again at about 65' from the uoi axis, indicating nonlinear concentrations. However, the minimum, as would be indicated by closed curves, seems to be absent (figure not shown). The lack of aclear minimum indicates that very steep gradients are not desirable for estimation of both aoi and b'i simultaneously, since steep gradients tend to reduce peak asymmetry. One parameter that the CIM isotherm has over the simple Langmuir isotherm is Z'i. For Zoi vs a'i, there is a trough about loo off from the aoi axis with very clear, closed curves in the center. For Z'i vs boi, a similar set of contours is seen. In both cases, Zoi is more clearly defined than a'i or boi since the trough of minimum values is closer to being parallel to the aoi or boi axis. The small range in the fitted values of Z"i for all gradient slopes (Table IV) supports the contour analysis and indicates that the value of Zoi can be estimated with certainty. Table IV also shows, however, a wide range of k h t estimates, which is supported by a contour plot of k h t vs a'i, which has a long trough running parallel to the k h t axis. A contour study of k h t vs boi also shows a very elongated trough parallel to the k h t axis. Estimation of the salt mass transfer coefficient is thus difficult without taking data on the exit concentration of the salt. However, the poor estimate of k h t does not affect the estimates of the isotherm parameters. The protein mass transfer parameter exhibited similar contour shapes against isotherm parameters as before for the simple Langmuir isotherm (Whitley et al., 1989b). As such, k~~ can be well estimated, even though its value can change with pulse size, pulse concentration, flow rate, and gradient conditions (see Tables I11 and IV). 4.3. Consideration of Salt Competition. The continuously changing salt concentration under gradient elution can produce unusual peak shapes for a variety of reasons. If the counterion concentration becomes high enough during the gradient, it could effectively compete with the solute, resulting in a lower loading for the solute. This effect can readily be represented by allowing each component in the CIM isotherm (eq 2) to have two different Z values, one for the aoi term and one for the bo;term. The resulting expression would then allow the capacity to
*%
5
0.2 0.15 0.1 0.05 0
5
0
0.35 0.3-l ... 0.25
10
1
c
0.2 0.1s 0.1 0.05 0 0
5
10
15
20
25
30
35
40
45
50
e
Figure 9. Experimental gradient elution curves (-) of BSA at pH 8.00 on Q-Sepharosefrom 0.157 to 0.248 N salt with gradient times (inmin) as noted. Solid lines are CIM with salt competition predictions (A) from the three isocratic pulse experiments, (B) from combined data of Figure 2, and (C) from direct fit of the 5-min gradient.
change as a function of salt concentration:
j-1
The inability of the CIM to represent all three isocratic runs in Figure 1with a single set of parameters is due to both a change in salt competition and the presence of multiple forms. By using eqs 12 and 13, rewritten for eq 14, the isotherm parameters shown in Table V were determined for the three isocratic experiments. The resulting predictions give only fair agreement with the isocratic elution data of Figure 1 (model curves not shown). This same set of isotherm parameters also gives only fair predictions of the gradient series (Figure 9A). Retention time is roughly predicted, but spreading is poorly described. If all the data shown in Figure 2 were used in estimating the gradient parameters in eq 14, a different set of parameters was obtained (see Table V). These values predict the five gradient elution experiments very well (Figure 9B) and do a reasonable job of predicting the three isocratic experiments (Figure 1, curves c). The close agreement is obtained because the isotherm parameters estimated from batch equilibrium data (Figure 2) represent average values if multiple forms are present. The average parameters can predict gradient behavior better than the isotherm parameters predicted from isocratic pulse elution, as will be discussed in the next section.
Biotechnol. hog., 1991, Vol. 7, No. 6
55 1
Table VI. Multiform Simulation Parameters for VERSE-LC fraction of total injected aoi 555.5
b'i,
6.0.5.5
ZOi
P i , 0
5
10
15
20
25
30
35
40
45
50
e Figure 10. Effect of salt competition on peak shape and elution for a 15-min salt gradient.
Upon fitting eq 14 to the 5-min gradient, we obtained a third set of parameter values (Table V), which agree fairly well with the CIM parameters determined for all the gradient data of Table IV. The fit matched the data closely for this isotherm (Figure 9C). These parameters are also able to give good prediction of the 1-, lo-, and 15-min gradient data. The fit for the 25-min gradient data was only marginally better than the single Zo CIM fit and was still unable to account for the secondary peak of of the 25-min gradient elution. Conformational changes or aggregation is the likely cause of these gradient parameter estimates differing from those from the isocratic estimates. Since the relative values of the two Z parameters in eq 14 can strongly affect retention behavior, a more detailed look at those parameters is needed. A series of simulations with various ratios of Zoa,i/Zob,i are shown in Figure 10 (ao = 0.0006, bo = 4.0, kLdt = 1.5, and klPmbin= 0.01). If Zoais greater than Zob, there will be a decrease in maximum solute capacity with increasing salt concentration. In Figure 10, however, the extreme tailing often seen in gradient elution of proteins (case of Z o a = 5.5, Zob = 6.5) results from increasing capacity with increasing salt concentration. Such a trend is not usually observed in ion-exchange systems. 4.4. Effects of MultipleConformationson Isocratic and Gradient Elution. A protein may exist as two or more forms due to denaturation or aggregation. In batch equilibration systems, such reactions are allowed to reach equilibrium, and the isotherm parameters estimated represent average values of multiple forms. In isocratic elution, if the reaction is much faster than the controlling mass transfer rate, a single (merged) peak will result. Isotherm parameters estimated from the merged peak represent average values of the individual forms (Whitley et al., 1991). If the reaction is much slower than the controlling mass transfer rate, the two forms will be well separated, exhibiting the individual retention properties of the two forms. If the reaction is on the same order of magnitude as the controlling mass transfer rate, the distribution of the two forms is continuously being shifted by the separation of the two forms during migration in a column, while reaction proceeds in an attempt to maintain the equilibrium distribution. For example, the ratio of the two peaks can change with flow rate or column length. Therefore, the interplay of reaction and separation results in complex elution behavior even for isocratic elution. In gradient elution, the behavior can be even more complex than in isocratic elution. Salt levels may affect affinity, salt competition, reaction rates, and the equilibrium distribution of various forms. Using VERSE-LC (Whitley, 1990;Whitley et al., 1991; Berninger et al., 1991), one can study the effects of conformational changes in isocratic or gradient elution. One can see that BSA seems to have a small, second peak for the 0.157 N salt level (Figure 1A). At a salt concentration of 0.207 N and higher the second peak becomes less
M-' cm2 min-1
Ep,,cm2 min-1
form AI 0.85 5.0 X lo4 2.4 5.73 3.564 x 10" 2.500 X 10-6
form A2 0.15 15.0 X lo-" 4.5 5.78 3.564 x 2.000 x 10-6
noticeable. In the various gradient systems, the shoulder is more pronounced for longer gradients. A longer gradient keeps the two peaks more separated and therefore more distant from equilibrium distribution than does a short gradient. The interconversion rate had to be slow relative to column residence time based on the partially separated peaks for the 0.157 N isocratic elution and for the 25-min gradient elution (Figure 7). Therefore, as a first approximation the BSA can be represented as two independent binding forms, denoted A1 and Az. Determination of isotherm parameters and distribution of BSA between the two forms was an iterative process. Single-form fitted isotherm parameters were used as a starting point. An initial guess of distribution was made and the 25-min gradient was simulated. Once that curve was well matched, isocratic simulations were conducted to compare effluent histories to isocratic data. On the basis of that comparison, parameter adjustments were made. Comparisons alternated between isocratic and gradient data until satisfactory agreement was achieved for both. With this single set of parameters (Table VI), VERSELC gave reasonable predictions of the 0.207 and 0.248 N isocratic elution experiments (curves not shown) and of all gradient durations (d curves of Figures 3-7). The predictions are much better than for just a singlecomponent prediction of one gradient slope by the parameters of another gradient slope. The single-solute parameters (Table IV for the gradient average fit) represent an average of the two-component simulation parameters (Table VI). Consideration of both salt competition and multiple forms may be necessary in modeling some protein separations. The 0.157 N isocratic elution shows even more extreme retention than can be accommodated in the model and still explain the other isocratic and gradient data. The longer retention time of BSA for the 0.157 N experiment could make any shift in the ratio of the two forms noticeable. As a test of these possibilities, an interconversion was set up in the model (eq 15) so that the k+
A, + A, k-
(15)
dimensionless reaction rate, @k+ [see Whitley et al. (1991) for a full discussion of this term], and equilibrium distribution, KD,could be adjusted until a good fit of the data was obtained. In Figure lA, curve f represents the VERSE-LC prediction for a2k+= 5.0 and KD = 0.40 (k+ = 5.0 min-l and 12- = 12.5 min-l). The isotherm and mass transfer parameters are unchanged from those of Table VI. By a shift in distribution from an averaged value of 0.1765 for gradient elution simulations to 0.40 at 0.157 N salt, the agreement with experimental data was much improved. At a low salt concentration there thus appears to be a significant shift in distribution to more highly retained forms. To summarize these results into an algorithm for parameter estimation, one should begin with a shallow gradient to see if multiple forms are present, as indicated by shouldering (Figure 7, for example). If more than one
Biotechnol. Prog., 1991, Vol. 7, No. 6
552
form is present, averaged isotherm parameters can be obtained under gradient elution conditions where the two peaks are merged. For a single component under salt gradient, one can subdivide the parameter estimation into two parts. First, aoiand Zoi can be estimated at low loading (small pulse volume and concentration) and an intermediate gradient duration. The value of boi is best estimated at high peak concentration and asymmetry. Shallow gradients give more peak asymmetry but a lower concentration, while steep gradients give more symmetric peaks but a higher concentration. Consequently, a large pulse and intermediate gradient steepness is best for a first run to estimate the value of boi. If variation of boiwith gradient duration is large, salt competition should be incorporated into the analysis. If aggregation or denaturation reactions exist and are ignored, isocratic pulse elution can give inaccurate isotherm parameters.
5. Conclusions A modified CIM isotherm which considers salt competition is able to predict BSA gradient elution from a mixture of batch equilibrium and some isocratic pulse elution data. The same isotherm was also able to predict elution for various gradient slopes when fitted to an intermediate slope gradient experiment. A series of isocratic pulse elution runs gave only fair prediction of gradient elution. The presence of multiple forms, whether due to denaturation or to aggregation, made the isocratic experiments poor predictors of gradient behavior. Isotherm parameters for multiple forms are readily averaged in batch experiments because of the long equilibration times. Similarly, gradient experiments yield averaged parameters because the salt gradient tends to merge the closely eluting forms. Slower flow rates in isocratic elution can help reduce the discrepancy by allowing forms to merge through interconversion. A detailed contour analysis on the CIM isotherm showed that the OBS SA, Z'BSA, and kLm parameters can be estimated with certainty from a single gradient elution peak. As an alternative to determining averaged parameters, consideration of two binding forms using VERSE-LC, an advanced rate model, gave good agreement with experimental data over the entire range of salt gradient durations. Multiple-form parameter determination is tedious for nonlinear isotherms, however. Consideration of salt competition, with the effects of multiple forms being averaged into the parameter estimates, is sufficient for the BSA system studied if one uses gradient elution data to predict gradient behavior.
Notation isotherm parameter primary form of BSA secondary form of BSA isotherm parameter, M(pore)-l concentration, M counterion modulator isotherm Brownian diffusivity, cm2 min-l effective intraparticle diffusivity, cm2min-l flow rate, mL min-1 reaction rate, min-l overall mass transfer coefficient, cm min-l equilibrium distribution column length, cm number of experimental points in fitting number of components number of stages particle radius, cm
t V
zo
time, min stage volume, mL ratio of eluent to salt valence
Greek Letters cb interparticle void fraction tP intraparticle porosity T percolation time, min $ modulator concentration, M +A2 defined by eq 11, M2 @.k+ Thiele modulus of reaction 6 dimensionless time Subscripts initial value b bulk phase exP experimental data i, i component counter model model prediction n stage counter P particle phase P pulse
0
Superscripts 0 CIM isotherm parameter solid phase
Acknowledgment This work was supported by the National Science Foundation under Grants BCS 8912150and CBT 8620221. Pharmacia LKB donated the Q-Sepharose Fast Flow exchanger gel and chromatography columns. Some computing resources were provided by the National Center for Supercomputing Applications through Grant CBT 900015N.
Literature Cited Alm, R. S. GradientElutionAnalysis. 11. Oligosaccharides.Acta Chem. Scand. 1952,6,1186-1193. Alm, R. S.; Williams, R. J. P.; Tiselius, A. Gradient Elution Analysis. I. A GeneralTreatment. Acta Chem. Scand. 1952, 6,826-836.
Antia, F. D.; Horvlth, Cs. Gradient Elution in Non-Linear Preparative LC. J. Chromatogr. 1989, 484, 1-27. Berninger, J. A.; Whitley, R. D.; Zhang, X.; Wang, N.-H. L. A Versatile Model for Simulation of Reaction and Non-Equilibrium Dynamics in Multicomponent Fixed-Bed Adsorption Processes. Comput. Chem. Eng. 1991, in press. Brown, J. M. Estimation of Protein Equilibrium Parameters Using Isocratic and Gradient Pulse Techniques. M.S. Thesis, Purdue University, West Lafayette, IN, 1990. Cen, P. L.; Yang, R. T. Analytic Solution for Adsorber Breakthrough Curves with Bidisperse Sorbents (Zeolites). AZChE J. 1986, 32 (lo), 1635-1641. Cowan, G. H.; Gosling, I. S.; Sweetenham, W. P. Modeling Methods to Aid the Design and Optimization of Batch Stirred Tank and Packed-Bed Column Adsorption and Chromatography Units. J . Chromatogr. 1989, 484, 187-210. Cox, G. B.; Antle, P. E.; Snyder,L. R. Preparative Separation of Peptide and Protein Samples by High-Performance Liquid Chromatography with Gradient Elution: 11. Experimental Examples Compared with Theory. J. Chromatogr. 1988,444, 325-344.
Czok, M.; Guiochon, G. Pitfalls in the Choice of Isotherms for the Calculation of Band Profiles in Preparative Chromatography. J. Chromatogr. 1991,537, 497-506. Eble, J. E.; Grob, R. L.; Antle, P. E.; Snyder, L. R. Simplified Description of High-PerformanceLiquid Chromatographic Separation Under Overload Conditions, Based on the Craig
553
Biotechml. Prog., 1991, Vol. 7, No. 6
Distribution Model. V. Gradient Elution Separation. J . Chromatogr. 1987,405,51-66. El Fallah, M. Z.; Guiochon, G. Comparison of Experimental and Calculated Results in Overloaded Gradient Elution Chromatography for a Single-Component Band. Anal. Chem. 1991, 63,859-867. Ghrist, B. F. D.; Snyder, L. R. Design of Optimized HighPerformance Liquid Chromatographic Gradients for the Separation of Either Small or Large Molecules: 11. Background and Theory. J . Chromatogr. 1988a,459,25-41. Ghrist, B. F. D.; Snyder, L. R. Design of Optimized HighPerformance Liquid Chromatographic Gradients for the Separation of Either Small or Large Molecules: 111. An Overall Strategy and Its Application to Several Examples. J. Chromatogr. 198813,459,43-63. Ghrist, B. F. D.; Cooperman, B. S.; Snyder, L. R. Design of Optimized High-Performance Liquid Chromatographic Gradients for the Separation of Either Small or Large Molecules: I. Minimizing Errors in Computer Simulations. J. Chromatogr. 1988,459,1-23. Goldberg, M. E. The Second Translation of the Genetic Message. Trends Biochem. Sci. 1985,10, 388-391. Gooding, K. M.; Schmuck, M. N. Comparison of Weak and Strong High-PerformanceAnion-Exchange Chromatography. J. Chromatogr. 1985,327,139-146. Hearn, M. T. W.; Anspach, B. Chemical, Physical, and Biochemical Concepts in Isolation and Purification of Proteins. In Separation Processes i n Biotechnology; Asenjo, J. A., Ed.; Marcel Dekker: New York, 1990;pp 17-64. Hearn, M. T. W.; Hodder, A. N.; Aguilar, M. I. High-Performance Liquid Chromatography of Amino Acids, Peptides and Proteins: LXXXVII. Comparison of Retention and Bandwidth Properties of Proteins Eluted by Gradient and Isocratic AnionExchange Chromatography. J . Chromatogr. 1988,458,2744. Hodder, A. N.; Aguilar, M. I.; Hearn, M. T. W. High-Performance Liquid Chromatography of Amino Acids, Peptides and Proteins: LXXXIX. The Influence of Different Displacer Salts on the Retention Properties of Proteins Separated by Gradient Anion-Exchange Chromatography. J. Chromatogr. 1989,476, 391-411. Hodder, A. N.; Aguilar, M. I.; Hearn, M. T. W. High-Performance Liquid Chromatography of Amino Acids, Peptides and Proteins: XCVIII. The Influence of Different Displacer Salts on the Bandwidth Properties of Proteins Separated by Gradient Elution Anion-Exchange Chromatography. J . Chromatogr. 1990,512,41-56. Hodges, R. S.; Parker, J. M. R.; Mant, C. T.; Sharma, R. R. Computer Simulation of High-Performance Liquid Chromatographic Separations of Peptide and Protein Digests for Development of Size-Exclusion, Ion-Exchange, and ReversedPhase Chromatographic Methods. J.Chromatogr. 1988,458, 147-167. Huang, J.-X.; Guiochon, G. Competitive Adsorption Behavior in HPLC: cis- and trans-Androsterone on Silica. J. Colloid Interface Sci. 1989,128 (2),577-591. Kang, K.; McCoy, B. J. Protein Separation by Ion Exchange Chromatography: A Model for Gradient Elution. Biotechnol. Bioeng. 1989,33,786-790. Lee, C. K.; Yu, Q.; Kim, S. U.; Wang, N.-H. L. Mass Transfer Effects in Isocratic Non-Linear Elution Chromatography. J. Chromatogr. 1989,484,29-59. Lee, C Y . ; Lazarus, L. H.; Kaplan, N. 0. Purification of Dehydrogenases and Kinases by Affinity Chromatography. In Proceedings of the Third Enzyme Engineering Conference; Pye, E. K.; Weetall, H. H., Eds.; Plenum Press: New York, 1978;pp 299-311. Lin, B.; Ma, Z.; Golshan-Shirazi, S.; Guiochon, G. Study of the Representation of Competitive Isotherms and of the Interaction Between Adsorption Isotherms. J. Chromatogr. 1989, 475,1-11.
Mhatre, R.; Krull, I. S.; Stuting, H. H. Determination of biopolymer (protein) molecular weights by gradient elution, reversedphase high-performance liquid chromatography with low-angle laser light scattering detection. J. Chromatogr. 1990,502,2146. Petzold, L. R., DASSL: A Differential/Algebraic System Solver. Lawrence Livermore National Laboratory: Livermore, CA, 1982. Pharmacia LKB Biotechnology. Products Catalog;Pharmacia: Piscataway, NJ, 1989. Rouhana, N. Gradient Elution of Biochemicals in Ion Exchange Chromatography. M.S. Thesis, Purdue University, West Lafayette, IN, 1990. Scatchard, G.; Scheinberg, I. H.; Armstrong, S. H., Jr. Physical Chemistry of Protein Solutions. IV. The Combination of Human Serum Albumin with Chloride Ion. J. Am. Chem. SOC.1950,72, 535-540. Schoenmakers, P. J.; Billiet, H. A. H.; Tijssen, R.; De Galan, L. Gradient Selection in Reversed-Phase Liquid Chromatography. J. Chromatogr. 1978,149,519-537. Snyder, L. R. Gradient Elution. High-Perform. Liq. Chromatogr. 1980,I , 207-316. Snyder, L. R.; Stadalius, M. A.; Quarry, M. A. Gradient Elution in Reversed-Phase HPLC Separation of Macromolecules. Anal. Chem. 1983,55,1412A-1430A. Snyder, L. R.; Cox, G. B.; Antle, P. E. Preparative Separation of Peptide and Protein Samples by High-Performance Liquid Chromatography with Gradient Elution: I. The Craig Model as a Basis for Computer Simulations. J. Chromatogr. 1988, 444,303-324. Snyder, L. R.; Dolan, J. W.; Cox, G. B. Preparative HighPerformance Liquid Chromatography Under Isocratic Conditions. J. Chromatogr. 1989,483,63-84. Stout, R. W.; Sivakoff,S. I.;Ricker, R. D.; Snyder, L. R. Separation of Proteins by Gradient Elution from Ion-Exchange Columns: Optimizing Experimental Conditions. J. Chromatogr. 1986, 353,439-463. Van Cott, K. E.; Whitley, R. D.; Wang, N.-H. L. Effects of Temperature and Flow Rate on Frontal and Elution Chromatography of Aggregating Systems. Sep. Technol. 1991,I, 142152. Whitley, R. D. Dynamics of Nonlinear Multicomponent Chromatography-Interplay of Mass Transfer, Intrinsic Sorption Kinetics, and Reaction. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1990. Whitley,R.D.; Wachter,R.;Liu,F.; Wang,N.-H.L.Ion-Exchange Equilibria of Lysozyme, Myoglobin, and Bovine Serum Albumin: Effective Valence and Exchanger Capacity. J. Chromatogr. 1989a,465,137-156. Whitley, R. D.; Brown, J. M.; Karajgikar, N. P.; Wang, N.-H. L. Determination of Ion Exchange Equilibrium Parameters of Amino Acid and Protein Systems by an Impulse Response Technique. J. Chromatogr. 1989b,483,263-287. Whitley, R. D.; Van Cott, K. E.; Berninger, J. A.; Wang, N.-H. L. Effects of Protein Aggregation in Isocratic Nonlinear Chromatography. AZChE J. 1991,37 (4)555-568. Yamamoto, S.;Nakanishi, K.; Matsuno, R.; Kamikubo, T. Ion Exchange Chromatography of Proteins-Prediction of Elution Curves and Operating Conditions. 11. Experimental Verification. Biotechnol. Bioeng. 1983a,25, 1373-1391. Yamamoto, S.; Nakanishi, K.; Matsuno, R.; Kamikubo, T. Ion Exchange Chromatography of Proteins-Prediction of Elution Curves and Operating Conditions. I. Theoretical Considerations. Biotechnol. Bioeng. 1983b,25,1465-1483. Yamamoto, S.; Nomura, M.; Sano, Y. Adsorption Chromatography of Proteins: Determination of Optimum Conditions. AZChE J . 1987,33(9),1426-1434. Yamamoto, S.;Nakanishi, K.; Matsuno, R. Ion Exchange Chromatography of Proteins; Marcel Dekker: New York, 1988. Accepted September 24, 1991.
