Ind. Eng. Chem. Res. 2001, 40, 377-383
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Specific and Nonspecific Adsorption in Affinity Chromatography. Part II. Kinetic and Mass Transfer Studies E. M. Martı´n del Valle and M. A. Gala´ n* Departamento de Ingenierı´a Quı´mica, Universidad de Salamanca, Plaza de los Caidos 1-5, 37008 Salamanca, Spain
The partition coefficient, adsorption equilibrium inside particles, adsorption kinetics, and mass transfer effects have been studied for asparaginase on Sepharose 4B activated with cyanogen bromide and with hexamethylenediamine and L-(+)-chlorosuccinamic acid as the spacer arm and ligand, respectively, for temperature and pH ranges of 298-302 K and 7.5-8.6, respectively, and for an ionic strength value of 0.05 M NaCl. A dynamic model has been developed to describe the adsorption. This model allowed us to obtain values of the diffusion coefficients and the forward surface interaction rate constants (De ≈ 10-8 cm2/s, k1 ≈ 10-4 mL mg-1 s-1). The forward surface interaction rate constant does not vary with the pH; however, an Arrhenius-type variation was observed with temperature. The values obtained for these constants range between 3.78 × 10-4 and 6.80 × 10-4 mL mg-1 s-1. The diffusion coefficient increases with pH (7.5-8.6). Additionally, within the temperature range studied, this coefficient was seen to follow Arrhenius-type behavior, allowing for a determination of the activation energies of the diffusion process for the different pH values (7.95-16.03 kcal/mol). Introduction Affinity chromatography is a powerful technique for the purification of biological macromolecules such as proteins and is a good example of a highly selective separation method.1 The purification operation includes adsorption, washing, elution, and regeneration steps. In the adsorption stage of the process, liquid containing the compound to be purified is placed in contact with the adsorbent, and provided that adsorption is sufficiently specific, only the required protein will be adsorbed.2 However, the optimization and scale-up of such affinity separation procedures require that the equilibrium and mass transfer characteristics be fully understood. The adsorption of a protein from the bulk solution onto a particle of adsorbent involves a number of discrete steps. These steps, all of which contribute resistance to mass transfer, include transfer from bulk liquid to the outer surface of the particle (film diffusion resistance), movement by diffusion into the pores of the particle (pore diffusion), and actual chemical interaction at the binding site (surface reaction resistance).2 Martı´n del Valle and Gala´n,3 studying the adsorption of asparaginase on Sepharose 4B activated with cyanogen bromide and with hexamethylenediamine and L-(+)chlorosuccinamic acid as the spacer arm and ligand, respectively, found that almost 1/3 of the enzyme is retained on the adsorbent through nonspecific sites. The aim of the present work was to gain insight into the concentration inside the pores, which may be different from that in solution if mass transfer resistance occurs, possibly changing the adsorption equilibrium. To study both effects for the above-described system, the partition coefficient was determined according to the * Author to whom correspondence should be addressed.
model developed by Taylor and Swaisgood,4 and a modified model for the dynamic adsorption of proteins on porous adsorbents, described by Mao et al.5-6 was applied. This model includes external and internal mass transfer resistance and kinetic and adsorption equilibrium constants. The experiments were performed in a batch reactor for ranges of pH and temperature of 7.5-8.6 and 298302 K, respectively, because, in these ranges, the enzyme is stable.7 The ionic strength was 0.05 M NaCl, and the stirring speed was 190 rpm, as used in previous experiments with this system.3 Materials Asparaginase (E.C. 3.5.1.1) and Sepharose 4B were purchased from Sigma Chemical Corporation. The other chemicals used, hexamethylenediamine, L-asparagine, D-asparagine, sodium chloride, cyanogen bromide, boric acid, and sodium tetraborate were obtained from Merck. All chemicals were of reagent grade. L-(+)-Chlorosuccinamic acid was obtained by a Walden conversion from D-asparagine, as described by Holmberg.8 The physical properties of this compound were determined by proton nuclear magnetic resonance spectroscopy (1H NMR) and by rotary power (+54), indicating a purity of 96%, with a synthesis yield of about 35%. Theory A modified model, from the theory developed by Mao et al.5-6 was used to describe the adsorption behavior of proteins in a finite bath. In this model, the adsorption process together with external and internal mass transfer are considered. The overall mass balance is given by
C + (1 - )PCP + (1 - )(1 - P)CEL ) [ + (1 - )P]CT (1)
10.1021/ie000402f CCC: $20.00 © 2001 American Chemical Society Published on Web 12/13/2000
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where CP is the adsorbate concentration in the pore fluid and P is the particle void fraction. C is the adsorbate concentration in the bulk of the liquid phase. CEL is the adsorbate concentration in the solid phase, and CT is the equivalent adsorbate concentration when the total amount of the adsorbate in the system is assumed to be only in the liquid phase. For the initial concentration, the value of CT is given by
per unit volume of the adsorbent particles and R0 is the radius of the particle. From eqs 7 and 8, the rate of change in the adsorbate concentration can be written as
dCEL ) aKe(C - Ci) dt
(9)
C0 + (1 - )PCP0 + (1 - )(1 - P)CEL0
1 1 1 ) + Ke Kf Ki
(10)
CT )
+ (1 - )P
(2)
For adsorption with fresh adsorbent particles, CEL0 ) 0, CP0 ) 0, and CT can be written as
CT )
C0 + (1 - )P
(3)
When the pore fluid is lumped with the bulk fluid, CP ) C, and eq 1 becomes
[ + (1 - )P]C + (1 - )(1 - P)CEL ) [ + (1 - )P]CT (4) However, the differential form of eq 4 is
dCEL dC + Rv )0 dt dt
and
(5)
Equation 10 clearly shows that the overall resistance to mass transfer is the sum of the resistance in the liquid film and the resistance in the pore fluid. Surface Interaction. The interaction between the adsorbate and the immobilized ligand at the internal particle surface can be described by the second-order reversible equation
dCEL ) k1[(CELm - CEL)Ci - K′dCEL] dt
where k1 is the forward interaction rate constant, CELm is the maximum adsorption capacity of the immobilized ligand, and K′d is the desorption equilibrium constant. Eliminating Ci, CEL, and their derivatives from eqs 1, 4, 8, and 11, the concentration equation can be written as
where
Rv )
1 - [ + (1 - )P] + (1 - )P
(6)
(
)
(12)
A RvCELm - CT + C
(13)
1 1 dC + ) (C - x1)(C - x2) M k1 dt
where
The mass transfer rate of the adsorbate from the bulk fluid to the internal particle surface can be expressed as a two-step process. First, the adsorbate diffuses through the boundary layer. Then, it diffuses through the pore fluid, which is stagnant, and finally, it is adsorbed on or interacts with the surface. Both mass transfer processes can be described by a linear force approximation, and hence, so can the overall mass transfer process. The mass transfer rate is then given by
M) and
A ) aKeRv
where N is the mass flux of the adsorbate into the particle, Kf is the liquid film mass transfer coefficient, Ki is the apparent pore fluid mass transfer coefficient, and Ke is the overall effective liquid-phase mass transfer coefficient. C* is the intermediate concentration of the adsorbate in the liquid phase at the external surface of the particles, and Ci is the intermediate concentration of the adsorbate in the liquid phase at the internal surface of the particles. Assuming that the volume of the liquid film is negligible and that there is no accumulation of the adsorbate in the pore fluid, the rate of change in the concentration of the adsorbate in the solid phase must then be equal to the rate of mass transfer. Hence
(8)
where the term a ) (3/R0) is the external surface area
(14)
x1 and x2 are the roots of the quadratic equation
N ) Kf(C - C*) ) Ki(C* - Ci) ) Ke(C - Ci) (7)
dCEL ) aKf(C - C*) ) aKi(C* - Ci) dt
(11)
C2 - BC - K′dCT ) 0
(15)
B ) CT - RvCELm - K′d
(16)
1 x1 ) [B + xB2 + 4K′dCT] 2
(17)
1 x2 ) [B - xB2 + 4K′dCT] 2
(18)
where
and
In the present case, because the matrix is made up of porous spheres of Sepharose 4B with an average diameter of 90 µm and a specific surface of 5 m2 mL-1, of which only 8 cm2 correspond to the external surface, there exists the possibility that part of enzyme might be retained inside the matrix and not really adsorbed, such that the adsorption equilibrium, K′d, in eq 11 must be related to the free enzyme inside the spheres,
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obtained by subtracting from the total concentration of enzyme CE0 added to the reactor the concentration of enzyme in solution, CβE. CTL is the total amount of ligand attached to the particles (which is also on the surface, although this amount is assumed be very small in comparison with the amount of ligand inside the pores). Experimental Section
Figure 1. Spherical porous particle with liquid phases R and β inside and outside pores.
which can be different from the concentration of enzyme in solution if there is internal or external mass transfer resistance. This effect can be studied through the partition coefficient, KP, which can be calculated using the model developed by Taylor and Swaisgood.4 The partition coefficient is defined as the ratio between the concentration of enzyme inside the pores, phase R, and the concentration of enzyme in solution, phase β (Figure 1). Accordingly, a reversible adsorption equilibrium is established between the enzyme adsorbed and that not adsorbed in phase R. This can be described by
ER + L a EL
(19)
The equilibrium dissociation constant K′d is defined as
K′d )
CRECL CREL
(20)
Taking into account that
CTL ) CL + CEL
(21)
CRTE ) CREL + CRE
(22)
KP )
CRE
(23)
CβE
substituting into eq 20, and making some rearrangements, the following equation is obtained:
CRTE CβE
- KP )
CTL K′d + KPCβE
(24)
where CRTE, CβE, and CTL are known because CβE is the concentration of enzyme in the bulk solution. The value of CRTE, the total concentration of enzyme in phase R, is
Partition Coefficient. Several experiments were performed to obtain the values of the partition coefficients and desorption equilibrium constants inside the particles for asparaginase in Sepharose 4B activated with CNBr and with hexamethylenediamine and L-(+)chlorosuccinamic acid as the spacer arm and ligand, respectively. For each pH value (7.5, 8.0, and 8.6) and temperature (298, 300, and 302 K), two runs were made at two different ligand concentrations (9.55-13.2 mg mL-1) for CE0 ) 1.25 × 10-3 mg mL-1 and I ) 0.05 M NaCl. From the experimental data thus obtained, the values of KP and K′d were calculated (Table 1). From this table, it can be observed that , when the pH was increased the partition coefficient, KP, increased because of an increase in the amount of enzyme retained inside the particles.3 Also, when the temperature was increased, desorption also increased and hence K′d increased; the amount of enzyme in solution, CβE, also increased, with the value of KP decreasing. For the optimal adsorption conditions, pH 8.6, 298 K, and 0.05 M NaCl, 75% of the enzyme is retained on activated Sepharose 4B with hexamethylenediamine and L-(+)-chlorosuccinamic as the spacer arm and ligand, respectively.3 The partition coefficient allowed us to determine that, of this percentage, 1/3 of the enzyme is in solution or linked by nonspecific bonds inside the particles, and 2/3 is linked by specific bonds. In addition, the partition coefficient is a parameter that permits us to separate the enzyme linked by nonspecific bonds to the enzyme in solution inside the particles. In agreement with Martı´n del Valle and Gala´n,3 28% of the enzyme was retained on activated Sepharose 4B with hexamethylenediamine for pH 8.6, 298 K, and 0.05 M NaCl. This percentage was attributed to nonspecific bonds. To verify this finding, the partition coefficient was also determined for asparaginase on the adsorbent described above under the optimal adsorption conditions, and a value of 0.31 was obtained. This information allowed us to determine the enzyme concentration in solution inside the particles, CRE, and the enzyme concentration adsorbed inside the particles, CREL. The latter corresponds to 2% of total initial enzyme concentration in solution and is the real percentage of enzyme linked by nonspecific bonds. The values of the adsorption constants corresponding to the equilibrium established with the enzyme inside the particles were compared (Table 2) with those obtained considering that all of the enzyme not in solution was adsorbed inside the particles.3 These data show that the adsorption equilibrium constants inside the particles are lower than those obtained previously under all pH and temperature conditions studied. This difference in the values shows that the enzyme inside
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Table 1. Partition Coefficients and Desorption Equilibrium Constants Inside Particles pH ) 7.5 T (K) KP K′d (mg mL-1)
298 0.448 0.243
300 0.416 0.270
pH ) 8.0 302 0.351 0.340
298 0.469 0.161
300 0.454 0.190
pH ) 8.6 302 0.399 0.250
298 1.39 0.139
300 1.15 0.154
302 0.759 0.201
Table 2. Partition Coefficients and Adsorption Equilibrium Constants Inside Particles and Adsorption Equilibrium Constants between Bulk Solution and Particle pH ) 7.5 T (K) KP Ka (mL mg-1) K′a (mL mg-1)
298 0.448 5.07 4.11
300 0.416 4.74 3.70
pH ) 8.0 302 0.351 3.87 2.94
298 0.469 6.71 6.21
the particles is partly in solution and partly adsorbed in equilibrium with enzyme in solution. From the data provided in Table 2, it can also be appreciated that the difference between the constants is directly related to the values of the partition coefficients; the higher the value of the coefficient, the greater the concentration of enzyme in solution inside the particles and, hence, the greater the difference between the values of the adsorption equilibrium constants. Forward Surface Interaction Rate Constant and Mass Transfer Coefficients. The values of K′d determined previously were introduced into the model, and eqs 11 and 12 were used to determine the kinetic and mass transfer coefficients. Also, eq 12 implies eq 10. In this equation, Ke is a function of K-1 f , the inverse of the external mass transfer coefficient. Kf is assumed to be very large when the system is run with a stirring speed of 190 rpm. Therefore, K-1 f is approximately zero, and
Ke ) Ki
(26)
To obtain profiles of the concentration of enzyme in solution vs time, several experiments were performed at different temperatures (298, 300, and 302 K) in a pH range of 7.5-8.6, with I ) 0.05 M NaCl and with a stirring speed of 190 rpm. To obtain such profiles, the same procedure as was previously described for preliminary runs3 to determine the effect of stirring speed was followed. Comparison of Model Prediction with Experimental Profiles. The predicted concentration-time profiles were compared with the experimental data; the results are shown in Figure 2a,b,c. In this figure, the points are experimental data, and the lines are the model predictions. These plots show that, once equilibrium has been reached, the model predicts that the concentration of enzyme in solution will be greater than that determined experimentally. The fact that the particles are highly porous means that part of the enzyme remains trapped inside the pores but without really being adsorbed, as reported above. The model used predicts the true concentration of enzyme specifically adsorbed inside the particles since the adsorption equilibrium constants used (eq 11) correspond to the equilibrium established inside the particles, between the enzyme in solution and that adsorbed, which was previously determined using the model described by Taylor and Swaisgood.4 This concentration of adsorbed enzyme varies with pH and with the working temperature. From Figure 2a,b,c, it can be
300 0.454 6.25 5.26
pH ) 8.6 302 0.399 4.92 4.00
298 1.39 21.27 7.19
300 1.15 16.39 6.28
302 0.759 9.90 4.97
seen that adsorption increases with pH and decreases with temperature within the time range studied. Both effects were previously noted by Martı´n del Valle and Gala´n.3 The data corresponding to the concentration-time profiles were correlated using the MATLAB computer program, according to eq 12, which evaluates the mass transfer coefficient inside the particles, Ki, and the rate constant, k1, for the different pH and temperature conditions studied. Once the apparent mass transfer coefficient inside the particles has been determined, it is possible to obtain the values of the diffusion coefficient, bearing in mind the relationship between the two quantities
Ki )
D τR0 e
(27)
where is porosity, τ is the tortuosity factor, and R0 is the radius of the adsorbent particle. In the adsorbent used here, the porosity had a value of 0.98, and the radius of the particle was 4.5 × 10-5 m. However, the tortuosity factor was an unknown value. To determine this parameter, the random pore model9 was applied; this permits an estimation of the value of the tortuosity coefficient from the porosity via
τ)
1
(28)
such that τ ) 1.02 and the /τ ratio has a value of approximately 1. With these data, using eq 27, it is possible to determine the diffusion coefficient of the system. These values, together with Ki and k1, are shown in Table 3. Forward Surface Interaction Rate Constant. The forward surface interaction rate constants obtained show that the adsorption rate does not change substantially when pH is varied in the range studied as the values of the constants were very similar, 3.78 × 10-4 and 6.80 × 10-4 mL mg-1 s-1, as seen in Figure 3. In the same figure, it can be seen that, for a given pH, the rate increases when temperature is increased. This can be explained by taking into account the fact that the relationship between the rate constant and temperature follows Arrhenius-type behavior. Diffusion Coefficients. Table 3 shows that the diffusion coefficient of asparaginase in Sepharose 4B activated with cyanogen bromide and with hexamethylenediamine and L-(+) chlorosuccinamic acid as the spacer arm and ligand, respectively, varies with both pH and temperature and that these values are similar,
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 381
Figure 2. Theoretical and experimental concentration curves for the adsorption of asparaginase on activated Sepharose 4Bhexamethylenediamine-L-(+)-chlorosuccinamic acid for I ) 0.05 M NaCl and (a) pH 7.5, (b) pH 8.0, and (c) pH 8.6. Table 3. Forward Surface Interaction Rate Constants, Apparent Pore Fluid Mass Transfer Coefficients and Diffusion Coefficients pH ) 7.5 T (K) Ki × 106 (m min-1) k1 × 104 (mL mg-1 s-1) De × 108 (cm2 s-1)
298 0.09 4.33 0.07
300 0.15 5.72 0.12
pH ) 8.0 302 0.32 6.80 0.25
De ≈ 10-8 cm2/s, to the value of the coefficient determined by Gala´n et al.10 for the same system using a dynamic method. They are also similar to the values determined by Horstmann and Chase.2 Effect of pH. For a fixed temperature value, the diffusion coefficient increases with pH in the range studied. Thus, the maximum values are seen for pH ) 8.6 (Figure 4). This type of behavior can be explained by considering that the process of protein adsorption depends on pH.11 The fact that the enzyme is a charged species and that the solvent has a dipolar character means that electrostatic interactions are involved in the process. pH ) 8.6 corresponds to the isoelectric point of the enzyme; at pH values lower than the isoelectric point, there is an excess of positive charge in the enzyme, such that the interactions with the solvent are greater because of the
298 0.11 3.94 0.09
300 0.20 5.27 0.16
pH ) 8.6 302 0.48 5.94 0.37
298 2.83 3.78 2.20
300 4.50 4.88 3.51
302 6.86 5.62 4.14
dipolar nature of the latter. This means that the diffusion rate will decrease as such interactions increase, explaining why the diffusion coefficient decreases with the drop in pH. Effect of Temperature. Diffusion coefficients in liquids are usually proportional to temperature. However, the adsorption of biomolecules in affinity chromatography follows a behavior like heterogeneous catalysis, in which there is a superficial diffusion, mainly due to the effect of the spacer arms and ligands. In this kind of activated process, the diffusion coefficient varies exponentially with temperature, and this effect of temperature is usually described by an Arrhenius-type expression.9,12 According to Oliveira et al.,12 the diffusion process varies with temperature, an Arrhenius-type dependence being seen between the diffusion coefficient and the
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Figure 3. Forward interaction rate constant vs pH for asparaginase in activated Sepharose 4B with hexamethylenediamine and L-(+)-chlorosuccinamic acid as the spacer arm and ligand, respectively, for different temperatures and I ) 0.05 M NaCl.
Figure 5. Natural logarithm of the diffusion coefficient vs 1/T (K-1) for asparaginase in activated Sepharose 4B with hexamethylenediamine and L-(+)-chlorosuccinamic acid as the spacer arm and ligand, respectively, for different pH values and I ) 0.05 M NaCl. Table 4. Diffusion-Activated Energy for Activated Sepharose 4B with Hexamethylenediamine and L-(+)-Chlorosuccinamic Acid as the Spacer Arm and Ligand, Respectively pH)7.5
pH)8.0
pH)8.6
16.03
18.56
7.95
E (kcal/mol)
Figure 4. Diffusion coefficient vs pH for asparaginase in activated Sepharose 4B with hexamethylenediamine and L-(+)-chlorosuccinamic acid as the spacer arm and ligand, respectively, for different temperatures and I ) 0.05 M NaCl.
temperature. This means that it is possible to determine the activation energy involved in the process, according to -E RT
De ) A e
(29)
Plotting ln De against the reciprocal of temperature affords straight lines (Figure 5), whose slopes correspond to the value of the activation energy divided by the gas constant. Table 4 shows the values of these energies corresponding to different experimental conditions of pH and temperature, and it also shows that the activation energy corresponding to pH ) 8.6 (the isoelectric point of the enzyme) has the minimum value (7.95 kcal/mol) because, as mentioned above, the electrostatic interactions are weaker for this pH value. The values of the activation energy found are similar to those previously reported for processes of protein adsorption.12 Conclusions From the experiments reported here, it can be concluded that, in the process of adsorption of asparaginase on activated Sepharose 4B with hexamethylenediamine as the spacer arm and L-(+)-chlorosuccinamic acid as the ligand, the adsorption constants corresponding to the equilibrium established inside the particles of adsorbent where the concentration of enzyme within the particles is not the same as that in the bulk solution (KA ) 2.94-7.29 mg mL-1) are lower than those
established when the concentration of enzyme is assumed to be constant and the same in both phases (KA ) 3.87-21.27 mg mL-1). This shows that part of the enzyme present inside the particles is in solution and part is adsorbed in equilibrium with the part in solution. The partition coefficient was also determined for asparaginase on activated Sepharose 4B with hexamethylenediamine at pH ) 8.6, T ) 298 K, and I ) 0.05 M NaCl, the optimal adsorption conditions. The value obtained, 0.31, allowed for a determination that the percentage of enzyme linked by nonspecific bonds is about 2%. A dynamic model has been developed to describe the adsorption of asparaginase on the adsorbent described; this allowed us to obtain the values of the diffusion coefficients and the forward surface interaction rate constants (De ≈ 10-8 cm2/s, k1 ≈ 10-4 mL mg-1 s-1). The forward surface interaction rate constant does not vary when the pH of the system is changed; however, an Arrhenius-type variation is observed with temperature. The values obtained for these constants range between 3.78 × 10-4 and 6.80 × 10-4 mL mg-1 s-1. The diffusion coefficient increases with increasing pH (7.5-8.6), the maximum value, De ) 4.14 × 10-8 cm2/s, corresponding to pH ) 8.6 (the isoelectric point of the enzyme) and T ) 302 K. Additionally, within the temperature range studied (298-302 K), this coefficient was seen to follow Arrhenius-type behavior, allowing for a determination of the activation energies of the diffusion process for the different pH values (7.95-16.03 kcal/mol). Acknowledgment This research was supported with funds from the Comision Interministerial de Ciencia y Tecnologı´a (CICYT). Ms. E.M. Martı´n del Valle gratefully acknowledges a fellowship from the same organization (CICYT). Notation a ) external surface area per unit volume of adsorbent particles, m
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 383 A ) parameter defined by eq 14 B ) parameter defined by eq 16 C ) adsorbate concentration in the liquid phase, mg mL-1 CREL ) concentration of enzyme adsorbed in the R phase, mg mL-1 R CTE ) total concentration of enzyme in the R phase, mg mL-1 C* ) intermediate adsorbate concentration in the liquid phase on external surface of particles, mg mL-1 C0 ) initial adsorbate concentration in the liquid phase, mg mL-1 CRE ) concentration of enzyme in the R phase, mg mL-1 CβE ) concentration of enzyme in the β phase, mg mL-1 CEL ) adsorbate concentration in the solid phase, mg mL-1 CELm ) maximum solid adsorption capacity, mg mL-1 Ci ) intermediate adsorbate concentration in the liquid phase on the internal surface of the particles, mg mL-1 CL ) concentration of ligand, mg mL-1 CP0 ) initial adsorbate concentration in the pore fluid, mg mL-1 CT ) equivalent adsorbate concentration when the total amount of adsorbate in the system was assumed to be in the liquid phase, mg mL-1 CTL ) total concentration of ligand attached to the particles, mg mL-1 De ) diffusion coefficient, cm2 s K′a ) enzyme-ligand adsorption equilibrium constant in the R phase, mL mg-1 K′d ) enzyme-ligand desorption equilibrium constant in the R phase, mg mL-1 k1 ) forward surface interaction rate constant, mL g-1 s-1 Ka ) enzyme-ligand adsorption equilibrium constant, mL mg-1 Kd ) enzyme-ligand desorption equilibrium constant, mg mL-1 Ke ) overall effective liquid-phase mass transfer coefficient, m s-1 Kf ) liquid-side film mass transfer coefficient, m s-1 Ki ) apparent pore fluid mass transfer coefficient, m s-1 KP ) partition coefficient, dimensionless N ) mass flux, m mg mL-1 s-1 Rv ) volume ratio of the solid phase to the liquid phase, dimensionless t ) time, min
x1 ) positive root of quadratic eq 15 x1 ) the other root of quadratic eq 15 Greek Letters ) volume fraction of the liquid phase in the finite bath, dimensionless p ) particle void fraction, dimensionless τ ) tortuosity factor, dimensionless
Literature Cited (1) Lowe, C. R.; Dean, P. D. G., Eds. Affinity Chromatography; Wiley-Interscience: London, 1974. (2) Horstmann, B. J.; Chase, H. A. Modelling the affinity adsorption of immunoblogulin G to protein A immobilised to agarose matrices. Chem. Eng. Res. Des. 1989, 67, 243-254. (3) Martı´n del Valle, E. M.; Gala´n, M. A. Specific and Nonspecific Adsorption in Affinity Chromatography. Part I. Preliminary and Equilibrium Studies. Ind. Eng. Chem. Res. 2001, 40, 369. (4) Taylor, B. J.; Swaisgood, H. E. A unified partition coefficient theory for chromatography, immobilized enzyme kinetics, and affinity chromatography. Biotechnol. Bioeng. 1981, 23, 1349-1366. (5) Mao, Q. M.; Stockmann, R.; Prince, I. G.; Hearn, M. T. W. Modelling of protein adsorption with nonporous and porous particles in a finite bath. J. Cromatogr. 1993, 646, 67-80. (6) Mao, Q. M.; Hearn, T. W. Optimization of affinity and ionexchange chromatographic process for the purification of proteins. Biotechnol. Bioeng. 1996, 52, 204-222. (7) Peter, P. K.; Milikin, B. E.; Bobbitt, J. L.; Grinnan, E. L.; Burck, P. J.; Frank, H. B.; Boeck, B.; Squires, R. W. Crystalline L-asparaginase from E. coli. J. Biol. Chem. 1970, 25, 3708-3715. (8) Holmberg, B. Stereochemische Studien. XIII: U ¨ ber β-Chlorsuccinamidsa¨uren. Chem. Ber. 1926, 59 (2), 1569-1580. (9) Smith, J. M. Chemical Engineering Kinetics; McGraw-Hill: New York, 1981. (10) Cantero, D.; Gil de Rebolen˜o, R.; Gala´n, M. A. Mass transport coefficients in gel permeation by moment analysis. Anal. Quı´m. 1987, 83, 641-646. (11) Bosma, J. C.; Wesslingh, J. A. pH dependence of ionexchange equilibrium of proteins. J. AIChE 1998, 44, 2399-2409. (12) Oliveira, J. C.; Sarmento, M. R.; Slatner, M.; Boulton, R. B. Kinetics of the adsorption of bovine serum albumin contained in a model wine solution by nonswelling ion-exchange resins. J. Food Eng. 1999, 39, 65-71.
Received for review April 11, 2000 Revised manuscript received July 31, 2000 Accepted September 20, 2000 IE000402F
