Determination of Langmuir Equilibrium Parameters of Bovine Serum Albumin from Batch Uptake Kinetics On-Line Number 9073 M

Determination of Langmuir Equilibrium Parameters of Bovine Serum Albumin from Batch Uptake Kinetics On-Line Number 9073 M. A. Hashim 1 and K. H. Chu 2 1 Department of Chemical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia, E-mail: [email protected] 2 Department of Chemical and Process Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, E-mail: [email protected]

ABSTRACT

The simple Langmuir isotherm model is often used to describe the equilibrium behavior of protein adsorption to a wide variety of adsorbents. Estimates of the two adjustable parameters of the Langmuir model, qm and Kd, are usually made by fitting the model equation to equilibrium data obtained from batch equilibration methods. In this work, the possibility of estimating qm and Kd for the adsorption of bovine serum albumin to a cation-exchanger from batch kinetic data was investigated. A rate model based on the kinetic form of the Langmuir isotherm with three adjustable parameters (qm, Kd, and a rate constant) was fitted to a single kinetic profile. The value of the saturation capacity qm identified by this approach was in very close agreement with the qm value determined from equilibrium data. However, the value of the dissociation constant Kd could not be retrieved from the kinetic profile as the model fit was insensitive to this parameter. Sensitivity analysis provided valuable insights into the identifiability of the three model parameters.

KEYWORDS Adsorption equilibrium, ion-exchange kinetics, Langmuir isotherm, parameter estimation

INTRODUCTION

Bioproducts such as proteins are usually obtained from fermentation processes. Extraction of the product from the fermentation broth invariably involves some form of chromatography (Lyddiatt, 2002). In particular, there has been extensive research on the use of ion-exchangers for the separation and purification of proteins. Currently, a large number of commercial cation and anion-exchangers are available for protein purification, typically prepared by coupling charged ligands to mechanically rigid porous beads (Chang and Lenhoff, 1998; DePhillips and Lenhoff, 2001; Staby and Jensen, 2001; Staby et al., 2000). Production-scale ion-exchange chromatography is usually conducted in fixed-bed columns packed with an appropriate ion-exchanger. However, assessing the performance of ion-exchange media is more often conducted in a batch configuration than a fixed-bed because it is simpler to model a batch apparatus. For example, experimental data generated from batch measurements can easily be quantified in terms of equilibrium and transport parameters which can then be incorporated within a larger process model to assess the behavior of a fixed-bed column. It is therefore not surprising that batch contacting procedures remain the method of choice for many researchers although dynamic techniques such as frontal analysis have been proposed for characterizing protein adsorption.

1 A body of work exists in the adsorption literature on the evaluation of equilibrium and mass transport parameters for various proteins in a wide variety of ion-exchange adsorbents using the batch approach (Bautista et al., 2000; Bosma and Wesselingh, 1998; Carta and Ubiera, 2003; Chen at al., 2002, 2003; Conder and Hayek, 2000; Garke et al., 1999; Habbaba and Ulgen, 1997; Hunter and Carta, 2000; Lan et al, 2001; Lewus et al., 1998; Lewus and Carta, 1999; Rowe et al., 1999; Soriano et al., 1999; Weaver and Carta, 1996; Wesselingh and Bosma, 2001; Wright et al., 1998; Yao et al., 2003; Zhou et al., 2002). This communication focuses on the modeling of protein adsorption equilibrium which is described quantitatively in terms of isotherm models. The most commonly used isotherm model is the Langmuir equation (Langmuir, 1918) with two adjustable parameters: the saturation capacity qm and the dissociation constant Kd. In most published studies these two Langmuir parameters were estimated by fitting the Langmuir equation or its linearized forms to equilibrium data generated by the batch equilibration method. Generally, the batch equilibration method requires extensive manual manipulation and is very time consuming. In this paper, we examine the possibility of identifying the two Langmuir parameters for protein adsorption to a cation-exchanger by matching uptake kinetic data from a batch apparatus to a rate model based on the kinetic form of the Langmuir isotherm. This approach offers a fast alternative to the conventional modeling approach based on batch equilibrium data. While batch kinetic measurements have been extensively used to estimate mass transport parameters, they have not been used to estimate equilibrium parameters.

THEORY

Consider a batch vessel which initially contains only a protein solution at a concentration ci. At time zero, fresh ion-exchange particles are added to the vessel. A material balance on the protein in the bulk liquid phase and the adsorbent phase is given by the following equation

dq dc v =−V (1) dt dt where q is the ion-exchange phase protein concentration, c is the liquid phase protein concentration, v is the ion-exchange adsorbent volume, V is the liquid volume, and t is the time. Adsorption of the protein to the ion-exchanger is assumed to be monovalent and homogeneous according to the following reversible reaction

PA+⋅U PA where P represents the protein molecule, A represents the adsorption site on the ion-exchanger, and PA⋅ is the protein-ion-exchanger complex. The rate of protein adsorption for an interaction described by the above reaction is given by

dq = k c(q − q) − k q (2) dt 1 m 2 where k1 is the forward interaction rate constant, k2 is the reverse interaction rate constant, and qm is the saturation capacity of the ion-exchanger. At equilibrium, Eq. (2) leads to the familiar Langmuir isotherm model

qmce qe = (3) K d + ce where the subscript e denotes an equilibrium value and Kd is the dissociation constant given by the following relation

k2 K d = (4) k1

Substituting Eq. (4) into Eq. (2) gives

dq = k c(q − q) − k K q (5) dt 1 m 1 d

Integrating Eqs. (1) and (5) with the appropriate initial conditions gives the following analytical solution (Horstmann et al., 1986)

⎡ v ⎤ (b + a) ⎢1 − exp( − 2a k1 t )⎥ c 1 v V = 1 − ⎣ ⎦ (6) ci ci V ⎡ (b + a) v ⎤ ⎢ − exp( − 2a k1 t )⎥ ⎣ (b − a ) V ⎦ where the parameters a and b are defined as

22 VVV⎛⎞ a =b−++ cqim ; b = 0.5⎜⎟ c i q m K d (6a) vvv⎝⎠

Eq. (6) is the solution of the rate model based on the kinetic form of the Langmuir isotherm from which the concentration-time profile for a given batch system can be calculated. When the experimental conditions are specified (ci, v, and V), Eq. (6) can be fitted to batch concentration-time data to identify the three adjustable parameters (k1, Kd, and qm) which appear nonlinearly in the equation.

MATERIALS AND METHODS

Materials

Bovine serum albumin (BSA) was purchased from Sigma. Acetic acid and sodium acetate were obtained from Fluka. The pH of an acetate buffer solution was maintained at 5. The cation-exchanger - Fractogel EMD SO3 650 (M) was obtained from Merck.

3 Batch Uptake Kinetics

The experimental apparatus is shown schematically in Figure 1. A vessel containing a protein solution at pH 5 was incubated and agitated in a shaking water bath maintained at 25 °C. A pump (Model 250, Perkin Elmer) was used to circulate the protein solution in a recycle loop. The cation-exchange particles were excluded from the recycle loop by a 10 µm HPLC pump inlet filter. The protein concentration in the liquid phase was monitored continuously using a UV/VIS spectrophotometer equipped with a flow cell (Lambda 3B, Perkin Elmer). Absorbance versus time data were recorded and converted into concentration versus time data by a PC. A typical kinetic experiment was carried out in the following manner. The initial absorbance of the protein solution in the vessel was measured, then at time zero, an ion-exchange suspension was added to the vessel. The absorbance decreased with time as the protein was being adsorbed, and levelled off as apparent equilibrium was reached. The experimental conditions are summarized in Table 1.

Adsorbent Table 1. Experimental conditions for batch Protein Solution kinetic experiment.

UV Detector ci (µmol/L) V (L) v (L) 22.5 5 x 10-2 2.5 x 10-4

Water Bath Shaker

Figure 1. Schematic of batch apparatus.

Batch Uptake Equilibrium

A series of protein solutions having different concentrations were placed in individual centrifuge tubes with the cation-exchanger. The tubes were rotated end-over-end at 25 °C for 24 hours. After equilibration, the slurry was centrifuged, and the protein concentration of the supernatant was measured by UV absorbance as described above. The amount of protein adsorbed to the cation-exchanger was calculated by mass balance.

Parameter Estimation

The best-fit values of the three adjustable parameters (k1, Kd, and qm) in Eq. (6) were estimated by minimizing the error between experimental transient uptake data and model calculations. The minimization algorithm is based on a combination of Gauss-Newton and Levenberg-Marquardt methods. The same nonlinear least squares curve fitting was used to identify qm and Kd in Eq. (3) by fitting the equation to measured equilibrium data.

4 RESULTS AND DISCUSSION

Although protein binding to adsorbent is a complex phenomenon, simple models such as the Langmuir isotherm are often used to describe the equilibrium characteristics of protein adsorption to a wide variety of adsorbents. The Langmuir model assumes that the adsorption process takes place on a surface composed of a fixed number of binding sites of equal energy, one molecule being adsorbed per binding site until monolayer coverage is achieved (Langmuir, 1918). Although these assumptions are often violated in protein adsorption, the Langmuir model gives a satisfactory correlation of equilibrium data in most situations. This is probably due to the fact that a mathematical model with two adjustable parameters is sufficient to account for the hyperbolic isotherm often encountered in protein adsorption. The Langmuir model therefore plays an important role in the domain of protein adsorption and estimation of its two parameters from experimental data is of practical interest. In this work, the possibility of retrieving the Langmuir parameters qm and Kd from a single kinetic profile obtained from a batch apparatus was investigated.

Batch Uptake Kinetics

Figure 2 shows batch uptake data for BSA on the cation-exchanger Fractogel EMD. The concentration decay profile of the solution phase is expressed in terms of normalized solution concentration (c/ci). As seen in this figure, the dynamic profile exhibits a slow approach towards equilibrium. The solid line in Figure 2 represents the fit obtained with the kinetic 1.0 model (Eq. (6)). It is evident that an excellent fit over the entire time course of protein uptake could be obtained. The best-fit values of the adjustable 0.9 i parameters in Eq. (6) are listed in Table 2. 0.8 c c / Table 2. Best-fit values of kinetic model parameters (Eq. (6)). 0.7

q (µmol/L) K (µmol/L) k (L/µmol.s) 0.6 m d 1 0 2000 4000 6000 8000 10000 1767 - 1.17 x 10-5 Time (s)

Table 2 shows that the saturation capacity of Figure 2. Concentration decay profile of BSA. Line: kinetic model fit, Eq. (6). the cation-exchanger (qm) for BSA is quite substantial. Since the pH of the experiment was set at 5, BSA with an isoelectric point of about 5 has a zero net charge. The fact that there is appreciable BSA uptake at pH 5 emphasizes that the ion-exchange mechanism relies on localized positive charges on the protein surface rather than the net charge of the protein (Levins and Ruckenstein, 1988). The forward rate constant k1 is an apparent rate constant which includes contributions from intrinsic adsorption kinetics, film diffusion, and intraparticle diffusion because the rate model as presented above is a simplified approach which lumps all kinetic and mass transport processes into a single rate constant. Unfortunately, a unique value of Kd could not be determined because the multi-parameter search algorithm returned the initial estimate of Kd over a wide range of initial values tested, indicating that the model was not at all sensitive to Kd. By contrast, the qm and k1 parameters always converged to the values listed in Table 2 regardless of the

5 starting guesses. Consequently, it is of paramount importance to systematically investigate the identifiability of the three parameters in Eq. (6) which has been shown to provide an excellent fit to the measured data shown in Figure 2.

Parameter Sensitivity Analysis

The preceding discussion indicates that it is difficult to simultaneously obtain unique estimates of qm, Kd, and k1 by fitting Eq. (6) to a single batch kinetic profile. In this section, we consider the ability of the multi-parameter search algorithm to obtain unique estimates of the individual parameters of the nonlinear rate model by conducting a parameter sensitivity analysis. Several approaches to parameter 0.35

5 5 sensitivity analysis are available (Beck and Arnold, 2 2 1 1 0.25 0 0 1977; Smith and Missen, 2003). One such approach 0 . 0 .

0 5

mol/L) 0 2 is based on analyzing the residual distribution µ

0 ( 5 d 0 plotted as contours (Whitley et al., 1989). The 0.15 2

6 K .

approach involves fixing all other parameters of a

0 multi-parameter model and systematically varying 0.05 pairs of two parameters to generate model responses. Corresponding residual distribution is then 1700 1720 1740 1760 1780 1800 1820 calculated by comparison with experimental data. In qm (µmol/L) this work the residual distribution is computed in Figure 3. Contours of qm vs Kd with terms of sum of squares of error (SSE) values: -5 constant k = 1.17x10 . 1 2 SSE = ∑[]()c ci exp − ()c ci model (7)

Figures 3 and 4 show the SSE values plotted as 0.35

contours for various combinations of Kd and qm and 8 0 of K and k , respectively. The ranges of these 0.25 0 d 1 0 . 0 parameters used in the generation of the contours are mol/L) µ

(

the best-fit values of q and k (see Table 2) 4% d m 1 ± 0.15 6 6

0 0

K 8

0 0 0 and a reference value of K (0.1 µmol/L) 400%. A 0 0 d ± 0

. .

0 0 .

0 common feature can be seen in these results. 0.05 Comparing Figures 3 and 4 clearly demonstrates that 1.12E-05 1.15E-05 1.18E-05 1.21E-05 the contours in both figures are aligned with the Kd axis. This feature implies that the SSE value does k1 (L/µmol.s) not change greatly when Kd is varied from 0.025 to Figure 4. Contours of k vs K with 0.4 mol/L (i.e., by a factor of 16) at a given q or k 1 d µ m 1 constant q = 1767. value. Under the given conditions the model is m clearly not sensitive to the value of Kd within the selected range of the parametric space. This is consistent with the curve-fitting exercise which returned the initial guess of Kd as the best estimate. The sensitivity analysis presented above reveals that without

6 considering parameter identifiability simple multi-parameter curve fitting methods may provide erroneous parameter values for nonlinear models. Finally, Figure 5 shows contours of k1 versus qm which are diagonal, indicating that the two parameters are somewhat correlated. Nevertheless, the innermost contour does close, enclosing a minimum SSE value. It may be concluded that both parameters may be estimated with a high degree of reliability. The accuracy of the qm estimate can be assessed by comparing it with the estimate retrieved from batch equilibrium data, as discussed below.

Batch Uptake Equilibrium 1.22E-05 0 .0 0 1.20E-05 00 .0 75 01 Uptake equilibrium of BSA in the cation- exchanger measured by the conventional batch 1.18E-05 0.0 equilibration method is shown in Figure 6. As mol.s) 0 00 µ .0 5 00 seen in this figure, the isotherm is nearly 1.16E-05 75

(L/ 1 rectangular, indicating highly favorable protein k 0 .0 1.14E-05 01 adsorption. The solid line in Figure 6 represents a description of the equilibrium behavior in 1.12E-05 terms of the Langmuir isotherm model (Eq. (3)). 1740 1750 1760 1770 1780 1790 1800 The two parameters in the Langmuir model qm qm (µmol/L) and Kd were obtained from a nonlinear fit of the Figure 5. Contours of q vs k with data and are listed in Table 3. As seen in Figure m 1 constant K = 0.1. 6, the agreement between calculated and d experimental data is excellent.

Table 3. Best-fit values of equilibrium model 2000 parameters (Eq. (3)). 1500

q (µmol/L) K (µmol/L) m d mol/L) 1000 µ

1732 0.19 ( e q 500 Comparison of Tables 2 and 3 indicates 0 that the resulting qm value is 2% smaller than the previous estimate identified from kinetic data. 0 102030405060 This excellent agreement confirms the accuracy ce (µmol/L) of the qm estimate obtained from batch uptake kinetics. However, as discussed above, the value Figure 6. Equilibrium isotherm of BSA. Line: equilibrium model fit, Eq. (3). of Kd could not be retrieved from the kinetic profile due to model insensitivity towards this parameter. In contrast, Kd can easily be identified from the equilibrium data obtained by the conventional batch equilibration method. However, due to the nearly rectangular shape of the isotherm estimation of Kd by nonlinear regression is very sensitive to the initial slope of the isotherm. Accurate data describing the ascending part of rectangular isotherms are not easy to obtain in practice because very low protein concentrations would have to be utilized in the batch equilibrium experiments. As a result, it may not be possible to retrieve unique estimates of Kd in practice although Kd is a priori identifiable from batch

7 equilibrium data. The issue of practical retrievability of Kd has been, for the most part, neglected in the field of isotherm determination by the batch equilibration method.

CONCLUSIONS

In this study, estimation of Langmuir equilibrium parameters from a single kinetic profile obtained from a batch apparatus was investigated. The saturation capacity of a cation-exchanger (qm) for bovine serum albumin identified by this approach was in very close agreement with the qm value retrieved from equilibrium data generated by the conventional batch equilibration method. Unique estimates of qm can therefore be retrieved from batch kinetic data with some confidence. However, sensitivity analysis showed that it was not possible to identify a unique value for the dissociation constant Kd under the given conditions. Insensitivity of the rate model to Kd may be due to either the experimental design (initial conditions of the batch apparatus) or the nature of the model itself. The influence of initial conditions (ci, V, and v) on the identifiability of Kd was not considered in this work and will be the subject of future communications.

ACKNOWLEDGEMENT

The authors thank Ms Grace Tsan (deceased) for laboratory assistance.

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