Optimization of Gradient Profiles in Ion-Exchange Chromatography for

Feb 3, 1997 - Gradient elution chromatography plays a very important role in the large-scale separation of biomolecules. The major problem with gradie...
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Ind. Eng. Chem. Res. 1997, 36, 444-450

Optimization of Gradient Profiles in Ion-Exchange Chromatography for Protein Purification Robert G. Luo Department of Chemical Engineering, Chemistry, and Environmental Science, New Jersey Institute of Technology, Newark, New Jersey 07102

James T. Hsu* Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, Pennsylvania 18015

Gradient elution chromatography plays a very important role in the large-scale separation of biomolecules. The major problem with gradient elution is that the development, scaleup, and optimization of the process depend on trial and error. This is usually time-consuming and expensive, especially for large-scale processes. Using chemical engineering concepts, the optimization of gradient elution chromatography consisted of two parts. In the first part, rate parameters and gradient correlations were determined according to experimental data from isocratic elution runs. In the second part, which is the main focus of this study, the optimization strategy of gradient elution chromatography based on the resolution optimization factor was developed. The resolution optimization factor considered not only the product purity and yield but also the operating time and product dilution. The methodology of optimizing gradient profiles for elution chromatography was demonstrated for protein purification. Introduction Gradient elution chromatography plays a very important role in the separation of biomolecules in both analytical studies and large-scale processes (Frey, 1990). In the authors’ previous studies, a mathematical model for gradient elution chromatography was developed and solved numerically (Luo and Hsu, 1992, 1993). This model and the computer programs were verified against experimental data and were used to estimate rate parameters and gradient correlations (Luo, 1994; Luo and Hsu, 1997). In this study, they will be used to optimize gradient elution processes according to preset control conditions. To optimize a gradient elution process, one first needs an objective function, which accurately defines the separation efficiency. Peak resolution is commonly used by most chromatographers to describe the degree of separation. In many bioseparations where operating time and product dilution are not as important as product purity and yield, resolution alone can be used as the objective function for gradient elution optimization. In the situations where operating time and product dilution have as great an impact on the cost basis as product purity and yield, resolution alone is not adequate to define the overall separation efficiency of a chromatographic process. In this paper, a new parameter called the “resolution optimization factor”, fo, which defines the separation efficiency of a chromatographic process, will be introduced. The resolution optimization factor is a function of both resolution and elution time. While resolution describes the degree of separation, the resolution optimization factor characterizes the separation efficiency. The utility of the resolution optimization factor will be demonstrated by applying it to case studies and comparing it with resolution. Later, the resolution optimization factor will be used, together with resolution, as an objective function for gradient elution optimization studies. * Author to whom all correspondence should be addressed. Phone: (610) 758-4257. Fax: (610) 758-5851. S0888-5885(96)00214-X CCC: $14.00

Figure 1. Effect of the mobile phase ionic strength, I, on adsorption equilibrium constants, Ka,i, of β-lactoglobulin A (LGA) and β-lactoglobulin B (LGB). Table 1. Parameters for Proteins in DEAE Sepharose CL-6B Column, pH ) 7.9 (Yamamoto et al., 1983) protein β-lactoglobulin A β-lactoglobulin B

DL/v (cm) 0.0128 0.0128

K0,i

Ai

Bi

0.6 0.6

2.57 × 9.41 × 10-5 10-4

-6.20 -6.28

In this work, a binary protein system was used for process optimization studies. For the given values of some control conditions, the computer program automatically searched for the optimal gradient slope that led to a desired resolution or a maximum resolution optimization factor. This paper presents a computational technique for selecting the best linear gradient (as measured at the column inlet) to achieve these goals. Model for Gradient Elution Chromatography The following mathematical model (in dimensionless form), which takes into account the combined effects of axial dispersion, film mass-transfer resistance, intraparticle diffusion, and surface adsorption kinetics, can be used to describe the gradient elution process in an ion-exchange column (Luo and Hsu, 1997). © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 445

Mass balance for sample components ∂Ui ∂2Ui ∂Ui 1 ) σp,iψiβi - σp,iψiβi 2 ∂τ Pe ∂X ∂X 3σp,iψiξi[Ui - Qi|η)1] (1) Ui(X>0,τ)0) ) Qi(X>0,τ)0) ) 0

(2)

∂Ui ∂X

(3)

|

X)0

) -Pe(Ui|X)0- - Ui|X)0+)

|

∂Ui ∂X p

)0

X)1

∂Qi ∂Si + ) pσp,i∇2Qi ∂τ ∂τ

Qi(η,τ)0) ) Si(η,τ)0) ) 0

|

1 ∂Qi Ka,i ∂η

∂Qi ∂η

η)1

|

η)0

)0

) ξi(Ui - Qi|η)1)

(4) (5) (6) (7) (8)

∂Si ) pω1,iQi - pω2,iSi ∂τ

(9)

Ui|X)0- ) 1 (τ0,i g τ g 0)

(10)

Ui|X)0- ) 0 (τ > τ0,i)

(11)

i ) 1, 2, ..., n. n is the number of sample components. For a linear adsorption isotherm, the equilibrium case of eq 9 can be written as

Si ) Ka,iQi Mass balance for the eluting buffer

[

(12)

]

∂Ub,j ∂2Ub,j ∂Ub,j 1 1 ) ψb,jβb,j ψ β b,j b,j ∂τ 1 + m/Kba,j Pe ∂X ∂X2 (13)

|

∂Ub,j ∂X

X)0

Ub,j(X>0,τ)0) ) 1

(14)

) -Pe[Ub,j|X)0- - Ub,j|X)0+]

(15)

|

∂Ub,j ∂X

X)1

)0

Ub,j|X)0- ) Ubinp,j(τ)

(16) (17)

j ) 1, 2, ..., h. h is the number of eluting buffer components. The definitions of the parameters are listed in the Nomenclature section. For ion-exchange chromatography with adsorption equilibrium, the relationship between equilibrium constants, Ka,i, and the ionic strength of the mobile phase, I, is expressed by

Ka,i ) K0,i + AiIBi

(18)

where K0,i are initial equilibrium constants and Ai and Bi are constants (Yamamoto et al., 1983; Luo and Hsu, 1997). In our work, a binary protein system of β-lactoglobulin A (LGA) and β-lactoglobulin B (LGB) was used. In eq 18, i ) 1 indicates LGA and i ) 2 indicates

LGB. Table 1 gives the values of K0,i, Ai, and Bi for both LGA and LGB. The decrease of Ka,i with an increase of the ionic strength, I, in the mobile phase for both proteins is shown in Figure 1. The above set of partial differential equations (PDEs) was then transformed into a set of ordinary differential equations (ODEs) by applying the method of orthogonal collocation (Finlayson, 1972, 1980; Raghavan and Ruthven, 1983; Villadsen and Stewart, 1967; Villadsen and Michelsen, 1978). FORTRAN programs were written to solve the ODEs by calling an ODE solver IVPAG with Gear’s method (Gear, 1971) in the IMSL software library (IMSL User’s Manual, 1989) on an IBM RS/6000 Model 950 mainframe computer. The effects of various parameters on sample elution peaks were investigated (Luo and Hsu, 1992, 1993). Rate parameters and gradient correlations were estimated and determined using experimental data and computer simulation results based on isocratic runs. The profiles of the eluting buffer’s ionic strength, which may be used to predict the operating conditions of gradient elution for separations of biomolecules, were also examined based on the model and compared to experimental results (Luo and Hsu, 1997). Definition of Resolution Resolution is defined as the retention time difference between two sample component peaks divided by the average base width of the two peaks (Jandera and Chracek, 1985):

Rs )

tr,2 - tr,1 1 (W + W2) 2 1

(19)

Here tr,1 is the retention time of the peak of component 1, and tr,2 is the retention time of the peak of component 2. W1 is the peak width of component 1, and W2 is the peak width of component 2. In many cases of bioseparations where operating time and product dilution are not as important as product purity and yield, a desired resolution can be used as the goal of gradient elution optimization. However, in the situations where operating time and product dilution have as great an impact on the cost basis as product purity and yield, resolution ignores another important factorsthe elution time. For example, if an isocratic elution is used with a mobile phase of weaker elution power, high resolutions between peaks can be achieved. However, the complete run will take a long time, and the peaks of the components with higher retention strengths will be very broad. Elution time is sometimes more important in preparative and large-scale chromatography, which can take hours to complete, than in analytical liquid chromatography. The long elution time may actually cause the eluting buffer or solvent to become an important impuritysa fact often ignored. Consequently, an extra step may be needed in the separation scheme to remove or reduce the amount of eluting buffer or solvent, which results in an increase in capital and operating costs and possible further loss of the yield. Therefore, resolution alone is not adequate to define the overall separation efficiency of a chromatographic process in the above situations. Another parameter is needed for the consideration of elution time and product dilution. Definitions of Resolution Optimization Factor In an isocratic elution run, a higher resolution can be achieved at the expense of a longer elution time. The

446 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997

Figure 2. Effect of the linear gradient slope, s, on the elution profiles of β-lactoglobulin A (LGA) and β-lactoglobulin B (LGB). Dp ) 1.68 × 10-5 cm2/min; d ) 0.011 cm.

Figure 3. Example illustrating the chromatographic resolution, Rs, of a binary system which can be changed at constant retention times, tr,1 and tr,2.

Table 2. Parameters of DEAE Sepharose CL-6B Column (Yamamoto et al., 1983) column diameter, D, cm column length, L, cm intraparticle void fraction,  initial ionic strength of the buffer, I0, M flow rate, F, mL/min sample loaded, SL, mL

1.5 10.0 0.35 0.11 0.3 6.0

situation is similar for a linear gradient elution process. Figure 2 shows the relative positions of proteins β-lactoglobulin A (LGA) peak and β-lactoglobulin B (LGB) peak under different values of the gradient slope, s. Here intraparticle diffusivity, Dp ) 1.68 × 10-5 cm2/ min, particle diameter, d ) 0.011 cm, and other parameters are given in Tables 1 and 2. It can be seen from Figure 2 that as s decreases from 0.1 to 0.005 M/mL, the separation between β-lactoglobulin A and β-lactoglobulin B becomes better. However, the elution times of the peaks with lower slopes (dotted peaks) are longer than those of the original peaks (solid peaks). That is, the lower the slope, the higher the resolution and the longer the elution time. If the slope decreases indefinitely, an isocratic elution will be reached with the highest resolution, but with the longest elution time and the most product dilution. The best kind of slope and corresponding resolution that are needed for a specific product depend on the comparison of the cost associated with the yield gained versus longer production times and the possibility of needing extra concentration steps due to product dilution. To evaluate these factors, another parameter, the average elution time, htr, is introduced:

ht r )

tr,1 + tr,2 2

(20)

htr represents the average elution time of the peaks under consideration. The overall separation efficiency of a chromatographic process cannot be measured if only one parameter, Rs or htr, is considered. For example, in a binary system, the peak resolution can be changed at constant retention times, as illustrated in Figure 3.

Figure 4. Effect of the linear gradient slope, s, on the resolution, Rs, and the dimensionless average elution time, T h r, of β-lactoglobulin A (LGA) and β-lactoglobulin B (LGB) separation. Table 3. Gradient Slope, s, and Corresponding Protein Resolution, Rs, Dimensionless Average Elution Time, T h r, and Optimization Factor, fo s (M/mL) Rs T hr fo

0.01 0.572 5.412 0.106

0.007 0.770 6.336 0.122

0.005 0.932 7.453 0.125

0.003 1.196 9.824 0.122

0.001 1.699 18.907 0.090

The changing values of Rs and T h r (T h r ) htrF/V0 is the dimensionless form of htr) with a decreasing gradient slope are listed in Table 3 and plotted in Figure 4. Here F is the flow rate, and V0 is the column void volume. It can be seen that when the slope decreases from 0.01 to 0.001 M/mL, the resolution increases from 0.572 to 1.699, and the average elution time also increases from 5.412 to 18.907. This result demonstrates that the slope affects both resolution and average elution time, which have opposite effects on the overall separation efficiency. A new parameter is needed to combine the effects of resolution and average elution time.

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 447

Local Resolution Optimization Factor, fo. By considering both resolution, Rs, and average elution time, htr, the measurement of the overall separation efficiency can be obtained. With this concept, the resolution optimization factor, fo, for a binary system is defined by

fo ) f(Rs,tr,i) )

Rs 1 (t + tr,2) 2 r,1

)

Rs ht r

(21)

Here fo is called the local resolution optimization factor because it only deals with two adjoining peaks next to each other. A resolution optimization factor for a multicomponent system will be called the overall resolution optimization factor. Here, the average elution time could be in its dimensional form htr or in its dimensionless form T h r. It should be pointed out that eq 21 is the form of the resolution optimization factor used in this study. The purpose in using this form is to demonstrate the concept of measuring the separation efficiency by considering both resolution and elution time. When applying this concept to different chromatographic systems, the form of the resolution optimization factor can be altered by using different equations or putting weights on resolution and average elution time. Overall Resolution Optimization Factor, Fo. For a multicomponent process, eq 21 can be modified to include resolutions of all neighboring peaks and the average elution time of these peaks:

Fo )

n

n

∑ n - 1i)1

Rs(1,2) + Rs(2,3) + ... + Rs(i,i+1)

× tr,1 + tr,2 + ... + tr,i + ... + Rs(n-1,n) + ... + tr,n

(22)

Here Fo is the overall resolution optimization factor, n is the number of sample components, Rs(i,i+1) is the resolution between the ith and the (i + 1)th peaks, and tr,i is the elution time of the ith peak. Applications of the Optimization Factor and Comparison with Resolution To demonstrate the utility of the resolution optimization factor, it was applied to the gradient elution process of β-lactoglobulin A and β-lactoglobulin B. In Table 3, the values of the resolution optimization factor are calculated according to eq 21 and listed as the fourth row in the table. In Figure 5, the resolution optimization factor, fo, and resolution, Rs, of β-lactoglobulin A and β-lactoglobulin B are plotted against the gradient slope, s. As s increases, unlike a constantly decreasing resolution, the optimization factor first increases and then decreases. Thus, there is a maximum value of fo at s ) 0.005 M/mL. This means that the gradient elution system will reach the highest separation efficiency when the gradient slope is set at 0.005 M/mL. The corresponding resolution is 0.932, which represents the degree of separation. Now two kinds of optimization goals are available for a gradient elution process. The first is a desired resolution; the second is a combination of resolution and resolution optimization factor. Which should be used mainly depends on the specific product and the separation process.

Figure 5. Effect of the linear gradient slope, s, on the resolution, Rs, and the resolution optimization factor, fo, of β-lactoglobulin A (LGA) and β-lactoglobulin B (LGB) separation.

Optimization Strategy In order to obtain rate parameters and gradient correlations, experiments with isocratic elution runs should be conducted prior to the optimization of gradient elution chromatography. These parameters and correlations can be estimated by comparing the experimental data with simulation results of an isocratic elution model (Luo, 1994; Luo and Hsu, 1997). They can then be put into the gradient elution model to optimize the separation process. Therefore, the optimization of gradient elution chromatography consists of two parts. The first part is rate parameter estimations and gradient correlation determinations. The second part is gradient profile optimization. The methodology employed in the algorithms in both parts is based on the optimization technique of Rosenbrock as modified by Swann (Fletcher, 1965). The control variable used in the present work is the mobile phase ionic strength as a function of time. The optimization strategy and algorithms can be extended to other types of variables such as pH or displacer concentration in a straightforward manner. Rate Parameter Estimations and Gradient Correlation Determinations. The first part of the optimization process is rate parameter estimations and gradient correlation determinations. The flow chart of this part is shown in Figure 6. The sum of the squared error, E, between the simulation data and the experimental data is used as a convergence criterion Np

Ek )

[Csimu(k,j) - Cexp(j)]2 ∑ j)1

(23)

where Csimu(j) represents simulated protein concentration, Cexp(j) represents experimental protein concentration, k is the iteration number, and Np is the number of experimental measured points. Process parameters such as the dispersion coefficient, DL, bed void fraction, , and intraparticle diffusivity, Dp, can be estimated from the experimental data and the model, as are the gradient correlations of adsorption

448 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997

Figure 6. Flow diagram for rate parameter estimation and gradient correlation determination.

equilibrium constants, Ka,i. The computing algorithm starts with reading experimental data, Texp(j) and Cexp(j), and then sets a control value of Eset. An isocratic elution model is used to simulate the chromatographic process. The simulation data are compared with the experimental data by using the sum of the squared error, E, as the convergence criterion. If Ek is less than or equal to Eset, then the rate parameters and the constants K0,i, Ai, and Bi in the correlations of Ka,i at a specific ionic strength, I, can be determined. Otherwise, the computer program estimates new values of the parameters and performs the same procedure until Ek is less than or equal to Eset. This procedure was described in detail by Luo and Hsu (1997). Gradient Profile Optimization. The second part is gradient profile optimization. The flow chart of this part is shown in Figure 7. A gradient elution model is used to simulate the chromatographic process, and k represents the iteration number. The computing algorithm starts by reading rate parameters and gradient correlations from the first part and then divides into path 1 and path 2. Path 1 (the left path in Figure 7) is for the situation where a certain value of resolution is required. In this case, a desired resolution, Rs,des, is set as the optimization goal. A linear gradient slope is set, and computer simulations are carried out. The resolutions between all adjoining sample component peaks are calculated from the simulation results. If any of the resolutions, Rs(i+1,i), are less than Rs,des, the program goes back to set a lower gradient slope and repeat the simulation until the desired resolution, Rs,des, is obtained and the corresponding gradient slope, s, is found. Figure 8 shows an example of this process where the desired resolution between β-lactoglobulin A and β-lactoglobulin B, Rs,des ) 1.2, is obtained by the computer and the corresponding gradient slope is 0.003 M/mL. Path 2 (the right path in Figure 7) is for the situation where the optimization goal is to reach a maximum resolution optimization factor, Fo,max, with the constraint that the resolution must be above a minimum value,

Figure 7. Flow diagram for gradient profile optimization.

Figure 8. Gradient elution optimization of β-lactoglobulin A and β-lactoglobulin B separation. Optimization goal: a desired resolution, Rs,des ) 1.2.

Rs,min. In this case, a value of Rs,min is first set in the program. Then a linear gradient slope is set. The resolutions between all adjoining sample component peaks and the objective function, Fo, are calculated from the simulation results. If any of the resolutions Rs(i+1,i) are less than Rs,min, the program goes back to set a lower gradient slope and runs again. Otherwise, it calculates Fo and compares the current Fo(k+1) with the previous value, Fo(k). If Fo(k+1) of the current iteration (k + 1) is greater than or equal to Fo(k) of the previous iteration, the program goes back to decrease the gradient slope and runs the procedure again. This continues until Fo reaches the maximum value, Fo,max. Figure 9 shows an example of this process, where a

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 449

Figure 9. Gradient elution optimization of β-lactoglobulin A and β-lactoglobulin B separation. Rs,min ) 0.77 is required. Optimization goal: a maximum resolution optimization factor.

minimum resolution between β-lactoglobulin A and β-lactoglobulin B was set at 0.77. The computer program first searched and obtained a slope, s ) 0.007 M/mL, which satisfies Rs,min, and then searched again to obtain a slope, s ) 0.005 M/mL, which gave the maximum resolution optimization factor, fo,max ) 0.125 with Rs > 0.77. Conclusions In this paper, the mathematical model and computer program developed in previous studies were used to carry out gradient elution optimization studies. For the purpose of defining the goal of gradient elution optimization, two cases were discussed. In the first case, where operating time and product dilution are not as important as product purity and yield, peak resolution alone can be used as the objective function for gradient elution optimization. In the second case, where operating time and product dilution are as important as product purity and yield on the cost basis of the product, resolution alone is not adequate to define the overall separation efficiency of a chromatographic process. A new parameter, the resolution optimization factor, was introduced to measure the separation efficiency of a chromatographic process. The resolution optimization factor was defined as the ratio of resolution to average elution time. It could be used as a complementary parameter to resolution in the gradient elution optimization studies. The utility of the resolution optimization factor was demonstrated. A strategy for gradient elution optimization was developed in this paper. This strategy consists of two parts. In the first part, process parameters are estimated and gradient correlations are determined from experimental data and simulation results with isocratic elution runs. In the second part, these parameters and correlations are put into the gradient elution model and used to carry out computer simulations. The computing algorithm has two paths. Path 1 is for the situation where a certain value of resolution is required. In this case, a value of Rs,des is set as the desired resolution, and the program automatically searches for a gradient

slope which results in a separation with Rs,des. Path 2 is for the situation where the optimization goal is to reach a maximum resolution optimization factor, Fo,max, under the condition that the resolution should be above a minimum value, Rs,min. In this case, a value of Rs,min is first set in the program. Then the program searches for a gradient slope which not only satisfies Rs,min but also results in a separation with a maximum resolution optimization factor. Using the mixture of β-lactoglobulin A and β-lactoglobulin B as a protein model system, both optimization paths were demonstrated. Although a linear gradient profile is used in this paper, the optimization of nonlinear gradient profiles can be carried out by adjusting the definition of the resolution optimization factor. This paper demonstrated that the gradient slope can be selected by varying the composition of the eluting buffer to optimize retention time for a given resolution. The optimization strategy and computer programs were developed to provide a methodology and an engineering tool for more efficient biochromatography development and operation. Which of the two computing algorithms described in part two should be chosen depends on the specific product, the chromatographic process, and the overall separation scheme. The best process can be selected by comparing the full cost of each alternative including their operating costs as well as any other costs should an extra concentration step be needed for one of the alternatives. Nomenclature Cb,j ) mobile phase concentration of buffer component j, kg/cm3 Cb0,j ) mobile phase initial concentration of buffer component j, kg/cm3 Cbinp,j ) mobile phase input concentration of buffer component j, kg/cm3 Cbp,j ) internal concentration of buffer component j in particles, kg/cm3 Cbs,j ) internal mobile phase surface concentration of buffer component j in particles, kg/cm3 Cexp ) experimental protein concentration, kg/cm3 Ci ) mobile phase sample concentration of component i, kg/cm3 Cm ) maximum sample concentration at column outlet, kg/ cm3 C0,i ) mobile phase sample input concentration of component i, kg/cm3 Cp,i ) internal sample fluid concentration of component i in particles, kg/cm3 Csimu ) simulated protein concentration, kg/cm3 Cs,i ) internal sample surface concentration of component i in particles, kg/cm3 ) D ) column diameter, cm DL ) axial dispersion coefficient, cm2/min Dp ) intraparticle diffusivity, cm2/min, Dp ) De/p d ) adsorbent particle diameter, cm E ) sum of the squared error Eset ) set limit for the sum of the squared error F ) flow rate, mL/min Fo ) overall resolution optimization factor, min-1 Fo,max ) maximum overall resolution optimization factor, min-1 fo ) local resolution optimization factor, min-1 fo,max ) maximum local resolution optimization factor, min-1 Iinp(t) ) input ionic strength profile I0 ) initial ionic strength ∆I ) change of ionic strength, M

450 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 Ka,i ) adsorption equilibrium constant of sample component i Kba,j ) adsorption equilibrium constant of buffer component j K0,i ) initial adsorption equilibrium constant of sample component i k ) iteration number kf ) film mass-transfer coefficient, cm/min k1,i ) forward adsorption kinetic constant of sample component i, 1/min k2,i ) backward adsorption kinetic constant of sample component i, 1/min L ) length of the adsorption bed, cm m ) /(1 - ) Np ) number of experimental measured points Pe ) VL/DL, Peclet number Qb,i ) Cbp,i/Cb0,i, dimensionless concentration of solvent component i in particles Qi ) Cp,i/C0,i, dimensionless fluid concentration of sample component i in particles R ) adsorbent particle radius, cm Rs ) resolution Rs,min ) minimum resolution Rs,des ) desired resolution Si ) Cs,i/C0,i, dimensionless surface concentration of sample component i in particles SL ) sample loaded, mL s ) ∆I/∆V, linear gradient slope, M/mL r ) radial distance from the center of the spherical particle, cm T ) tF/V0, dimensionless time, equivalent to θ used by Yamamoto et al. (1983b) T h r ) htrF/V0, dimensionless average retention time, min t ) time, min to,i ) input time of sample component i, min tr,i ) retention time of sample component i, min htr ) (thr,i + htr,i+1)/2, average retention time, min Ub,j ) Cb,j/Cb0,j, dimensionless mobile phase concentration of buffer component j Ubinp,j ) Cbinp,j/Cb0,j, dimensionless mobile phase input concentration of buffer component j Ui ) Ci/C0,i, dimensionless concentration of sample component i in mobile phase V ) elution volume, mL V0 ) column void volume, mL ∆V ) change of elution volume, mL v ) intersticial velocity, cm/min Wi ) peak width of sample component i, min X ) z/L, dimensionless axial distance z ) axial distance coordinate, cm Greek Letters  ) void fraction of the adsorption bed p ) pore porosity of the particle ∇2 ) Laplacian operator η ) r/R, dimensionless radial distance in the particle σp,i ) Dp,1/Dp,i, dimensionless diffusivity parameter for sample component i ω1,i ) k1,iR2/(pDp,1), dimensionless kinetic parameter for sample component i ω2,i ) k2,iR2/(pDp,1), dimensionless kinetic parameter for sample component i

ψi ) Ka,i(1 - )/, distribution ratio for sample component i ψb,j ) Kba,j(1 - )/, distribution ratio for buffer component j βi ) vR2/(LDp,iKa,i(1 - )), bed length parameters for sample component i βb,j ) vR2/(LDbp,jKba,j(1 - )), bed length parameters for buffer component j ξi ) kfR/(Dp,iKa,i), film resistance parameter for sample component i τ ) Dp,1t/R2, contact time parameter τ0,i ) Dp,it0,i/R2, input contact time parameter of sample component i

Literature Cited Finlayson, B. A. The Method of Weighted Residuals and Variational Principles; Academic Press: New York, 1972. Finlayson, B. A. Nonlinear Analysis in Chemical Engineering; McGraw-Hill, Inc.: New York, 1980. Fletcher, R. Function Evaluation Without Evaluating Derivatives: A Review. Comput. J. 1965, 8, 33. Frey, D. D. Asymptotic Relations for Preparative Gradient Elution Chromatography of Biomolecules. Biotechnol. Bioeng. 1990, 35, 1055. Gear, C. W. Numerical Initial-Value Problems in Ordinary Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1971. IMSL User’s Manual, IMSL Math Library, Fortran Subroutines for Mathematical Applications; IMSL: Houston, TX, 1989. Jandera, P.; Churacek, J. Gradient Elution in Column Liquid Chromatography; Elsevier: Amsterdam, The Netherlands, 1985. Luo, R. G. Engineering Studies of Gradient Elution Chromatography for Biomolecule Separations. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1994. Luo, R. G.; Hsu, J. T. Simulation of Gradient Elution Chromatography by the Method of Orthogonal Collocation. Proc. Summer Comput. Simul. Conf. 1992, 24, 668. Luo, R. G.; Hsu, J. T. Intraparticle Protein Diffusion Effect on Gradient Elution Chromatography. Sep. Technol. 1993, 3, 221. Luo, R. G.; Hsu, J. T. Experimental Determination of Rate Parameters and Gradient Correlations for Gradient Elution Chromatography. AIChE J. 1997, in press. Raghavan, N. S.; Ruthven, D. M. Numerical Simulation of a FixedBed Adsorption Column by the Method of Orthogonal Collocation. AIChE J. 1983, 29, 922. Villadsen, J. V.; Stewart, W. E. Solution of Boundary-Value Problems by Orthogonal Collocation. Chem. Eng. Sci. 1967, 22, 1483. Villadsen, J. V.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1978. Yamamoto, S.; Nakanishi, K. R.; Matsuno, R.; Kamikubo, T. Ion Exchange Chromatography of ProteinssPrediction of Elution Curves and Operating Conditions. II. Experimental Verification. Biotechnol. Bioeng. 1983, 25, 1373.

Received for review April 15, 1996 Revised manuscript received November 15, 1996 Accepted November 17, 1996X IE960214I

X Abstract published in Advance ACS Abstracts, January 1, 1997.