Hydrophobic interaction chromatography of proteins IV

Journal of Chromatography A, 1139 (2007) 84–94

Hydrophobic interaction chromatography of proteins IV Kinetics of protein spreading Emmerich Haimer, Anne Tscheliessnig, Rainer Hahn, Alois Jungbauer ∗ Department of Biotechnology, University of Natural Resources and Applied Life Sciences, Muthgasse 18, A-1190 Vienna, Austria Received 1 May 2006; received in revised form 24 October 2006; accepted 1 November 2006 Available online 20 November 2006

Abstract Adsorption of proteins on surfaces of hydrophobic interaction chromatography media is at least a two-stage process. Application of pure protein pulses (bovine serum albumin and ␤-lactoglobulin) to hydrophobic interaction chromatography media yielded two chromatographic peaks at low salt concentrations. At these salt concentrations, the adsorption process is affected by a second reaction, which can be interpreted as protein spreading or partial unfolding of the protein. The kinetic constants of the spreading reaction were derived from pulse response experiments at different residence times and varying concentrations by applying a modified adsorption model considering conformational changes. The obtained parameters were used to calculate uptake and breakthrough curves for spreading proteins. Although these parameters were determined at low saturation of the column, predictions of overloaded situations could match the experimental runs satisfactorily. Our findings suggest that proteins which are sensitive to conformational changes should be loaded at high salt concentrations in order to accelerate the adsorption reaction and to obtain steeper breakthrough curves. © 2006 Elsevier B.V. All rights reserved. Keywords: Hydrophobic interaction chromatography; Adsorption; Protein spreading

1. Introduction Hjerten [1] introduced the term hydrophobic interaction chromatography (HIC) previously described by Shaltiel as hydrophobic chromatography [2,3]. The method is often used as an orthogonal tool for protein purification [4,5] or even for matrix assisted protein refolding [6–8]. HIC may be used to separate proteins that cannot be separated by common chromatographic techniques like ion-exchange chromatography and size-exclusion chromatography. Additionally, it is an ideal subsequent step after precipitation with ammonium sulphate or after ion-exchange chromatography when proteins are eluted at high salt concentration [9]. Hydrophobic interactions that control selectivity are affected by a number of process variables, including temperature, pH, salt concentration and type, additives and chromatographic ligand type and density [10–16] Interaction of proteins with the hydrophobic ligand is highly selective [17–19].

∗

Corresponding author. Fax: +43 1 3697615. E-mail address: [email protected] (A. Jungbauer).

0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.11.003

However, its complex selectivity behaviour and potentially destabilizing effects on proteins limit application of HIC especially at very high feed concentrations. Conformational changes and yield losses induced by contact with the hydrophobic surfaces in HIC are disadvantages of the method [20]. It is known that proteins undergo conformational changes when they encounter a strongly hydrophobic surface as used in reversedphase liquid chromatography (RPLC) [21]. The unfolding reaction may be influenced by several process parameters [22–24]. Even for HIC, a more gentle method of protein separation, conformational changes have been reported [20,25–27]. Two chromatographic peaks were found for pure protein samples. The earlier eluting peak was thought to be native or predominately native protein, while the more retained, more hydrophobic species was presumed to be partially unfolded protein. Different techniques have been applied to study protein unfolding upon adsorption. Wu et al. [26,27] used an on-line photodiode array to look at absorbency ratios of chromatographic peaks. Raman spectroscopy and hydrogen-deuterium isotope exchange (HX) methods have been utilized to derive detailed information about the unfolding reactions of proteins

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

on hydrophobic surfaces [20,28]. Application of these methods clearly demonstrated that proteins might partially unfold upon contact with a hydrophobic surface. Extend of unfolding is highly dependent on the individual protein. Conformational changes of the protein may lead to decreased production yield. This may either be due to incorrect refolding following elution [29] or to capacity losses due to the increased molecular volume of the unfolded protein. Under certain conditions shallow elution fronts were observed resulting in low dynamic capacity. By applying the mathematical model proposed by Lundstrom [30] for modelling the breakthrough curves, it is possible to trace this elution fronts back to conformational changes of the protein [31]. In earlier work, pulse response experiments of pure protein samples were presented [25]. These chromatograms showed two discrete peaks. The second peak eluted as soon as the salt concentration was lowered. This peak was assumed to be the fraction of the loaded protein, which underwent at least partial conformational interconversion. Further experiments demonstrated that not every protein undergoes conformational changes upon adsorption. Furthermore, binding strength of the unfolded fraction depends on the individual protein. Injection of some pure protein samples onto a HIC column resulted in one peak, eluting in presence of (NH4 )2 SO4 . Others resulted in two peaks, both eluting in presence of the salt. Further ones resulted in two peaks, one eluting in presence of the salt and one immediately after changing to regeneration buffer. In the third case the unfolding also termed spreading reaction was irreversible in presence of salt. For an irreversible reaction, the spreading model reduces to three kinetic parameters, adsorption rate constant of the native fraction, desorption rate constant of the native fraction and spreading rate constant. Pulse response experiments at varying flow rates or pulse concentration were carried out and an attempt was made to predict the breakthrough behaviour from the pulse response data taking into account the spreading of the molecule. 2. Theory 2.1. Adsorption kinetics Adsorption of proteins is often described by the Langmuir isotherm, which is usually given by the following relationship: q = qm

Ka c 1 + Ka c

(1)

where q is the amount of bound protein, c is the bulk protein concentration, and qm is the maximal binding capacity. Ka is the adsorption constant which describes the ratio of the adsorption rate constant k1 to the desorption rate constant k−1 . The Langmuir kinetic considers protein concentration and amount of free attachment sites as parameters for description of adsorption. dq = k1 c(qm − q) − k−1 q dt

(2)

Based upon this model, Lundstrom proposed an extended model considering protein unfolding after adsorption [24,31]. A reaction scheme for this model is given in Fig. 1. The adsorption

85

Fig. 1. Schematical description of adsorption and spreading of proteins. For irreversible spreading, the system reduces to three reaction constants (k1 , k−1 , k2 ).

process can be described by the following correlations: dq1 = k1 c(qm − q1 − βq2 ) − k−1 q1 − k2 q1 (qm − q1 − βq2 ) dt (3) dq2 = k2 q1 (qm − q1 − βq2 ) dt

(4)

dq dq1 dq2 = + dt dt dt

(5)

were q1 is the amount of bound protein in the native state, q2 is the amount of bound protein in the denatured state, k1 and k−1 are the adsorption and desorption rate constants and k2 is the unfolding rate constant. c is the bulk protein concentration. β is a geometrical factor describing the ratio of surface area used in denatured and native state. Proteins are bound to a surface with an adsorption rate constant k1 . If free surface is available adsorbed particles can spread (unfold) according to the spreading reaction rate constant k2 . The spreading reaction may thus be limited by high adsorption rate constants. As the surface is fully occupied by protein in native state no further spreading reaction would be possible [24] except the spread species occupies the same space (β = 1). 2.2. Adsorption in fixed bed In order to model adsorption processes on the column, the spreading model was coupled with the mass balance, describing the transport within the column. The transport in the bulk phase is given by the following partial differential equation and the corresponding initial and boundary conditions: ∂c (1 − ε) ∂q ∂c ∂2 c + + ν = DL 2 ∂t ε ∂t ∂z ∂z

(6)

t=0

c = c(0, z)

(6a)

z=0

∂c v (c − cf (t)) = ∂z DL

(6b)

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z=L

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

∂c =0 ∂z

(6c)

with c as the bulk fluid concentration, cf (t) as the time-dependent feed concentration and DL as the axial dispersion coefficient. L is the column length and ε is the void fraction, which describes the relation of the extra particle volume of the mobile phase to the column volume. v is the interstitial fluid velocity. ∂q/∂t is described by the spreading model (Eqs. (3)–(5)). The model was solved numerically by the finite elements method. 3. Methodology 3.1. Buffers, proteins and stationary phase All buffer ingredients and bovine serum albumin (BSA) were purchased from Merck (Vienna, Austria). Lysozyme and ␤lactoglobulin were acquired from Sigma–Aldrich (Vienna, Austria). Toyopearl Butyl 650 M was purchased from Tosoh Haas (Japan). As loading buffer a 20 mM sodium phosphate buffer, pH 7.2 was prepared and ammonium sulphate was added to a final concentration of 0.7 m. As elution buffer a 20 mM sodium phosphate buffer, pH 7.2 was used. 3.2. Column experiments ¨ All experiments were performed on an AKTA Explorer 100 system (GE Healthcare, Sweden) consisting of a separation unit and a personal computer running the UNICORN 3.1 control system. 8 ml of Toyopearl Butyl 650 M were packed into a HR 10/10 column (I.D. 1 cm) from GE Healthcare and 2 ml into a HR 5/10 column (I.D. 0.5 cm). For the pulse response experiments, the column was equilibrated with 3 column volumes (CVs) loading buffer. 250 ␮l sample of varying protein concentration were injected and after elution with 5 CVs elution buffer regenerated with 4 CVs loading buffer. For studying kinetics of the unfolding process, the linear velocity was varied between 50 and 500 cm/h. For the breakthrough curve experiments, the column was equilibrated with 2 CVs loading buffer. Then 400 ml of protein solution in loading buffer with a concentration of 1 mg/ml was applied, bound protein eluted with 6 CVs elution buffer and regenerated with 4 CVs loading buffer. 3.3. Uptake kinetics Uptake kinetics were measured using a dip probe-method as described by Tscheliessnig et al. [32]. Experiments were performed on a system consisting of a dip probe (FDP7UV2002-ME, Avantes, Eerbeek, The Netherlands), a light source (AvaLight-DHS Deuterium Halogen Light Source, Avantes) and a spectrometer (AvaSpec-2048, Avantes). Thirty millilitres of protein solutions of different concentrations were provided in a vessel and stirred with a hanging magnetic stirrer at 300–450 rpm. 0.5 ml of the equilibrated chromatography medium were added and decreasing protein concentration was

continuously monitored employing absorbance measurement at 280 nm with the dip probe. The signal was transferred from the spectrometer to the computer using AVASOFT 6.0 (Avantes). 4. Results and discussion Understanding the mechanism of protein adsorption on HIC media is still challenging. While the underlying adsorption reaction in common methods, such as ion exchange or affinity chromatography may be assumed as single-stage reaction, HIC adsorption seems to incorporate more stages indicated by the observation of incomplete elution in pulse response experiments [14,25]. In order to study the kinetics of the proposed spreading process pulse response experiments were performed at different flow rates (Fig. 2). The retention time of the native fraction was equal to the contact time of the protein on the hydrophobic surface. The amount of spread protein (q2 ) was estimated by integration of the second peak, eluting during the regeneration phase. The native fraction (q1 ) was estimated by integration of the first peak. We assume that the second fraction of protein underwent a certain conformational change, because the binding strength to the surface is higher. In this paper we exclude surface heterogeneity and assume a homogenous surface for the native fraction on the fraction having partially changed the conformation. There is also enough evidence that the protein solution is homogenous prior to injection. The unfolding reaction rate constant k2 can be derived by assuming that the maximal binding capacity qm , as defined in Eq. (4), exceeds the actual surface coverage by the native and spreaded protein. Thus, we can write qm  q1 + βq2 and Eq. (4) can be simplified to dq2 = k2 q1 q∗ dt

(7)

where q* replaces the term (qm −q1 −βq2 ), the amount of protein loaded by the pulse divided by the column volume. Integration of Eq. (7)  t  q2 ∗ dq2 = k2 q1 q dt (8) 0

0

leads to q2 = k2 q1 q∗ t

(9)

and the relationship between bound protein in native and partially unfolded conformation q2 = k2 q∗ t (10) q1 The unfolding reaction rate constant k2 can be derived by assuming that the maximal binding capacity qm , as defined in Eq. (4), exceed the actual surface coverage by the native and spreaded protein qm  q1 + βq2 . When small protein pulses are applied, adsorption takes place in the linear range of the isotherm thus the maximal adsorbable

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

87

Fig. 2. Pulse response experiments of BSA (A) and ␤-lactoglobulin (C) on Toyopearl Butyl 650 M at varying flow rates and evaluation of the isocratic BSA (B) and ␤-lactoglobulin (D) pulses. Plots of q2 /q1 versus qxt with slope k2 .

amount of protein is equivalent to the applied pulse and the term (qm −q1 −βq2 ) is equal to q* which is determined by the amount of the applied protein pulse divided by the column volume. Plots of q2 /q1 versus q* t are given in Fig. 2. Linear regression of these plots resulted in a straight line with slope k2 [ml mg−1 s−1 ]. The unfolding reaction rate constant was estimated for Toyopearl Butyl 650 M (average particle diameter 65 ␮m) and Toyopearl Butyl 650 S (average particle diameter 35 ␮m). Both materials showed comparable unfolding reaction rate constants, 2.5 × 10−2 and 3.1 × 10−2 ml mg−1 s−1 for BSA on Toyopearl S type and M type, respectively. For ␤lactoglobulin a k2 of 6.9 × 10−3 and 4.1 × 10−3 ml mg−1 s−1 was determined. This implies that mass transfer does not superimpose the unfolding reaction, since mass transfer depends quadratically on the particle size. Furthermore, this assumes that the rate constants for adsorption and unfolding are small enough to apply a reaction model [33–35]. Further pulse response experiments were performed at different protein concentrations (Fig. 3). According to the spreading model the unfolding reaction is limited by the accessible surface area. With increasing surface coverage, available surface for spreading of adsorbed protein decreases. Consequently the

amount of unfolded protein (q2 ) should decrease with increasing sample concentration (c0 ). Plots of q2 /q* versus q* are shown in Fig. 3. For ␤-lactoglobulin (Fig. 3D), the consequences of limited surface area for protein spreading can be experimentally verified. BSA did not show a pronounced concentration dependent conformational change (Fig. 3B). This indicates a β value close to unity for BSA. As the required surface does not increase significantly during conformational interconversion upon adsorption the process would not be limited by increasing q* . Additionally, pulse response experiments with varying concentration were used to describe the adsorption behaviour of the native state. By applying a protein pulse to the column a certain fraction of the protein adsorbs to the surface and changes its conformation. The pulse migrates through the column without any change of dq* /dt, but the adsorption equilibrium is in competition with the spreading reaction. Changes in the native state result in changes in the unfolded state. Regarding the assumptions for estimation of k2 the term (qm −q1 −βq2 ) can be rewritten in terms of the amount of applied protein q* . Eq. (3) can be expressed as k1 cq∗ − k−1 q1 = k2 q1 q∗

(11)

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Fig. 3. Pulse response experiments for BSA (A) and ␤-lactoglobulin (C) at varying protein concentrations. BSA (B) and ␤-lactoglobulin (D) spreading in dependency of protein concentrations. For β > 1 increasing pulse concentrations lead to decreasing protein spreading due to sterical limitations.

in matrix form   k1 ∗ = [k2 q1 q∗ ] [cq q1 ] k−1

(12)

Solving this overdetermined system of equations yields the adsorption and desorption rate constants. Table 1 contains the estimated reaction parameters. Uptake curves were calculated solving numerically Eqs. (3)–(5) using the estimated reaction rate constants. The bulk protein concentration c(t) decreased during the batch uptake process. This fact was considered in the calculation by fitting the protein uptake to the empirical function c(t) = a0 + a1 t + a2 t 1.5 + a3 t 0.5

(13)

where c(t) is the time-dependent concentration and t is the time. The concentration c(t) was calculated for every time step in the numerical solution. Plots of the experimental and calculated uptake curves are given in Fig. 4. The β factor could be estimated by the uptake curves since all reaction rate constants could be determined by pulse response experiments. As expected BSA had a β value of approximately equal to 1 while for ␤-lactoglobulin a β value of approximately 3.5 was derived (Table 1). This can be interpreted that ␤lactoglobulin underwent substantial spreading upon adsorption on the stationary phase while BSA did not change its molecular volume significantly. Breakthrough curves (BTCs) for the investigated proteins and media are shown in Fig. 5. It can be seen, that the BTC for ␤lactoglobulin is steep and saturation is asymptotically reached

Table 1 Model parameters for Eqs. (3)–(5). Protein BSA ␤-Lactoglobulin

qm [mg/ml]

k1 [ml mg−1 s−1 ]

k−1 [s−1 ]

k2 [ml mg−1 s−1 ]

β

7.93 1.35

7.3 × 10−3

2.3 × 10−1

3.1 × 10−2

1 3(±1)

4.1 × 10−3

2.6 × 10−2

4.1 × 10−3

Parameter k2 was calculated from isocratic pulse response data performed at different residence times and applying Eq. (9). Parameters k1 , k−1 were calculated from isocratic pulse response data performed at different protein concentrations and applying Eq. (9). BSA and ␤-lactoglobulin. q was determined from batch uptake experiments.

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

Fig. 4. Comparison of calculated and measured uptake curve for BSA (A) and ␤-lactoglobulin (B) at 0.7 m (NH4 )2 SO4 . Uptake was measured by the dip probe method. Grey dots indicate dip probe data [22].

with a long tailing, while the BTC for BSA exhibits a very shallow shape. Breakthrough curves were predicted using Eqs. (3)–(6). The parameters used for the prediction were estimated by independent experiments, pulse response and batch uptake, respectively. The predicted BTCs show good agreement to the experimental data whereupon the β value for ␤-lactoglobulin had to be reduced to 2.5 in order to approximate the data. Additionally, the capacity of the fixed bed was somewhat higher as determined by the batch uptake method. For ␤-lactoglobulin the difference in qm between fixed bed and batch was 1.2 mg/ml, for BSA the difference was 1.5 mg/ml. The impact of the spreading reaction on the shape of the BTC has been theoretically demonstrated (Fig. 6). For low salt concentrations, the reaction rate constants for adsorption and spreading are within the same order or magnitude. Thus, the spreading reaction affects the adsorption equilibrium. Fig. 6A shows the effects of the unfolding rate constant (k2 ) on the BTC shape. It can be seen, that the spreading reaction leads to shallow breakthrough curves when the reaction rate constants are in the same order of magnitude. An increase of the β value compensates this effect. Thus, acceleration of the adsorption reaction will lead to steeper elution fronts. Faster adsorption reaction is usually

89

Fig. 5. Calculated (◦ ) and measured (-) BTC for ␤-lactoglobulin (A) and BSA (B) at 0.7 m (NH4 )2 SO4 .

obtained at high salt concentrations. This effect was experimentally confirmed (Fig. 7). The BTC of lysozyme is given as an example for a protein, which does not spread upon adsorption. It can be seen, that the BTCs for ␤-lactoglobulin and BSA deviate significantly from the BTC of lysozyme. Decoupling adsorption from the spreading reaction by increasing the salt concentration leads to equally sharp elution fronts of spreading and non-spreading proteins. Thus, a critical salt concentration exists, where shallow BTC can be observed. The same effect is evident for uptake curves. Fig. 6B shows the effects of k2 and the β value on the uptake curves. An increase of the adsorption reaction rate constant is necessary to sharpen the uptake curve. Thus, the system is expected to equilibrate earlier at higher adsorption reaction rate constants. This effect can be also observed in an uptake experiment (Fig. 8A). The equilibrium is reached earlier with increased salt concentration. Thus, increasing the salt concentration always increases HIC adsorption. The salt dependent change of the uptake curves is not a proof but a further indication of our hypothesis that spreading of protein occurs model during adsorption on HIC surfaces. As the β value for BSA is about 1 the spreading reaction dominates the shape of the uptake curve for 0.7 m (NH4 )2 SO4 . For ␤-lactoglobulin these trends were less visible (Fig. 8B) thus the β value counteracts the steepness of the uptake curve.

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E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

Fig. 6. (A) Effects of the spreading rate constant and the beta value on the elution front shape and (B) on the uptake curve.

Fig. 7. (A) BTC for spreading (BSA, ␤-lactoglobulin) and non-spreading (lysozyme) proteins in the region of same order or magnitude for adsorption and spreading reaction – the critical salt concentration and (B) below the critical salt concentration.

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

91

Fig. 8. Uptake curves for BSA (A) and ␤-lactoglobulin (B) at 0.7 m (NH4 )2 SO4 and 1.2 m (NH4 )2 SO4 (grey dots indicate dip probe data/lines are smoothed data).

Fig. 9 shows the experimental uptake of the two model proteins. The individual contributions calculated by the spreading model using the parameter obtained by pulse response experiments are superimposed to the overall uptake. For BSA almost

all protein bound to the HIC surface has undergone at least partial conformational change. In a theoretical analysis we assumed a larger k1 value by 2 orders of magnitude implicating faster adsorption kinetics. Despite the fast adsorption the

Fig. 9. Experimental and uptake curves for BSA (A) and ␤-lactoglobulin (C). Total native and spreaded protein has been calculated and superimposed. In (B), the k1 value has arbitrarily increased by a factor of 100 and in (D) ␤ was reduced from 3.8 to 1.

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E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

native fraction of bound protein disappears rapidly (Fig. 9B). For ␤-lactoglobulin a similar observation was made. In this case, the native fraction is larger. It has to be noted that the absolute binding capacity is very low at 0.7 molal ammonium sulphate. The theoretical analysis of spreading by reducing the β value to 1 shows that the capacity would double. This can be interpreted as follows. Preventing spreading would help to increase capacity. Several authors and in particular Jennissen [36–38] formulating the concept of critical hydrophobicity claiming to adsorb the protein at lowest possible salt concentration. Our results suggest to refine this strategy for certain proteins. If the respective proteins are available in pure state basic pulse response experiments can be performed in order to determine an operational range of base matrix and ammonium sulphate concentration where the adsorption step is not hindered by the spreading reaction. At the current stage, it is not clear if our calculations reflect real reactions occurring at the surface of an HIC medium. Possibly our model must be corrected in order to match the real situation. In case in situ measuring of unfolding on the surface is possible the evidence reported here would suggest a new design strategy for HIC of proteins. 5. Conclusion It is known, that proteins change conformation upon adsorption to surfaces [39–42]. This effect can be a problem in chromatographic purification steps. Several publications report of two peaks as a result of a pure protein pulse [7,12]. An analytical proof is very complex due to fast renaturation after desorption of the spreaded fraction. Therefore, another experimental approach to monitor additional reactions during HIC adsorption would be necessary. The typical two-peak chromatograms were used to estimate reaction constants for the hypothetical second reaction. Using these reaction constants it was possible to calculate breakthrough curves, which were similar to the experimentally determined data. In this way it was possible to explain the typical chromatograms by a spreading reaction. The results imply that HIC adsorption has to be considered as multiple-stage reaction with at least two stages, namely adsorption and spreading. The spreading reaction influences the adsorption process and leads to flatter uptake and breakthrough curves. Thus, the HIC adsorption process should be performed at high salt concentrations in order to yield high adsorption rate constants. By increasing the adsorption rate constant the adsorption process can be decoupled from protein spreading and remains the dominating process. This would suggest a new paradigm for design of HIC running conditions. Nomenclature

c DL F k1 k−1 k2

bulk protein concentration [mg/ml] axial dispersion coefficient [cm2 /s] fractional saturation [–] adsorption rate constant [ml mg−1 s−1 ] desorption rate constant [s−1 ] unfolding rate constant [ml mg−1 s−1 ]

adsorption constant [mg/ml] total protein (q1 + q2 ) [mg/ml] bound protein in native state [mg/ml] bound protein in partially unfolded state [mg/ml] static capacity [mg/ml] amount of protein in a pulse divided by column volume [mg/ml] bed axial coordinate [m]

Ka q q1 q2 qm q* z

Greek letters β geometrical factor (radius of denatured protein/radius of native protein) [–] ε void fraction [–] v interstitial velocity [cm/s] Appendix A A Langmuir adsorption with subsequent spreading (Eqs. (3)–(5)) and a mass balance for the transport in the bulk fluid phase (Eq. (6)) have been introduced to describe the adsorption behaviour of proteins on a hydrophobic surface. Combination of these equations resulted in an equation system with three unknowns, c, q1 and q2 , consisting of two ordinary differential equations (ODEs) and one partial differential equation (PDE) with the corresponding initial and boundary conditions.   ∂c (1 − ε) ∂q1 ∂2 c ∂q2 ∂c (A1-1a) + + ν = DL 2 + ε ∂t ∂t ∂z ∂z ∂t t=0

c = c(0, z)

(A1-1b)

z=0

∂c v = (c − cf (t)) ∂z DL

(A1-1c)

z=L

∂c =0 ∂z

(A1-1d)

dq1 = k1 c(qm − q1 − βq2 ) − k−1 q1 − k2 q1 (qm − q1 − βq2 ) dt (A1-2) dq2 = k2 q1 (qm − q1 − βq2 ) (A1-3) dt The numerical solution of this model was obtained by discretisation of the space coordinate using the Garlekin weighted residual method and subsequent integration of the resulting system of ODEs using an implemented ODE-solver in Matlab. The discretisation was performed as follows. The column length is divided by an arbitrary number, Ne, of finite elements with a constant width, dx, of all elements. The unknowns of the equation system are replaced by an approximating function, the so called trial function, c ∼ cˆ =

Ne+1 

cj ϕ j

(A1-4a)

j=1

q1 ∼ qˆ 1 =

Ne+1 

q1,j ϕj

j=1

(A1-4b)

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

AI1

93

∂q1 = k1 AI1 cj qm − k1 AI2 cj (q1,j + βq2,j ) − k−1 AI1 q1,j ∂t − k2 AI1 qj qm + k2 AI2 qj (q1,j + βq2,j )

AI1

∂q2 = k2 AI1 qj qm − k2 AI2 qj (q1,j + βq2,j ) ∂t

with AI1 =

AI2 =

q2 ∼ qˆ 2 =

q2,j ϕj

(A1-4c)

j=1

with ϕj as the base function. After introduction of the approximating functions (Eq. (A1-4)) into the equation system of the model (Eqs. (A1-1) to (A1-3)) the resulting residual is minimised using weighting functions. In case of the Garlekin weighted residual method, the weighting functions are the base functions used in the approximating functions (Eq. (A1-4)). The partial derivative of second order at the right hand side of Eq. (A1-1a) can be simplified applying integration by parts. The base functions ϕ were chosen to be quadratic functions thus each is defined over two elements (Fig. A.1). Each base function is equal unity at the corresponding node and zero at all others. After introducing a local coordinate system, with −1 ≤ ξ ≤ 1, the quadratic base functions are given as ϕ−1 (ξ) = − 21 ξ(1 − ξ)

(A1-5a)

ϕ0 (ξ) = 1 − ξ

(A1-5b)

ϕ−1 (ξ) =

2

1 2 ξ(1 + ξ)

(A1-5c)

(A1-6)

with M as any of the element’s matrices and I ∈ {1. . .Ne} the complete ODE system can be set up. Written in form of matrices for one element Eqs. (A1-1a)–(A1-3) are given as   ∂c 1 − ε I ∂q1 ∂q2 AI1 + A1 + ∂t ε ∂t ∂t = −vBI cj − DL EI c − vF I c + vF0 cfeed

1

−1



ϕi (ξ)ϕj (ξ)

dz L dξ = dξ Ne

4 15 ⎜ 2 ⎝ 15 1 − 15

(A1-7c)

1 − 15

2 15 16 15 2 15

2 15 4 15

⎛ 1

−1

ϕi (ξ)ϕi (ξ)ϕj (ξ)

dz L dξ = dξ Ne

⎟ ⎠ (A1-8a) 1 − 70

8 105 32 35 8 105

13 70 ⎜ 8 ⎝ 105 1 − 70

⎞

8 105 13 70

⎞ ⎟ ⎠

(A1-8b)

BI =

⎛



1

−1

− 21

∂ϕi (ξ) dξ dz ⎜ ϕj (ξ) dξ = ⎝ − 23 ∂ξ dz dξ 1 6

2 3

− 16

0

2 3 1 2

− 23

⎞ ⎟ ⎠ (A1-8c)

EI =



⎛ 1

−1

7 6

− 43

1 6

16 6 − 43

∂ϕi (ξ) dξ ∂ϕj (ξ) dξ dz Ne ⎜ 4 dξ = ⎝−3 ∂ξ dz ∂ξ dz dξ L

1 6

⎞

⎟ − 43 ⎠ 7 6

(A1-8d) and F as zero matrix of size Ne × Ne with F1,1 = 1 and F0 as F0(1,1) = 1. Arrangement of the submatricces according to the global coordinates (see Eq. (A1-6)) will give a set of 3 ×(Ne + 1) ODEs which can be solved in MatLab employing an ODE-solver, such as ode23t. References

The advantage of the local coordinates is that Eqs. (A11a)–(A1-3) need to be solved for only one subset of elements and by arrangement of the matrices of the elements applying I+1 I M(I−1)·3+3,(I−1)·3+3 = M3,3 + M1,1

⎛



Fig. A.1. Quadratic base functions defined in the local (x) and global (z) coordinates.

Ne+1 

(A1-7b)

(A1-7a)

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. Hjerten, J. Chromatogr. 87 (1973) 325. S. Shaltiel, Proc. Natl. Acad. Sci. USA 52 (1973) 430. S. Shaltiel, Proc. Natl. Acad. Sci. USA 52 (1973) 778. N. Jakobsson, M. Degerman, B. Nilsson, J. Chromatogr. A 1099 (2005) 157. S.M. Cramer, G. Jayaraman, Curr. Opin. Biotech. 4 (1993) 217. J.-J. Li, M. Venkataramana, A.-Q. Wang, S. Sanyal, J.-C. Janson, Z.-G. Su, Prot. Expr. Pur. 40 (2005) 327. J.-J. Li, Y.-D. Liu, F.-W. Wang, G.-H. Ma, Z.-G. Su, J. Chromatogr. A 1061 (2004) 193. A. Jungbauer, W. Kaar, R. Schlegl, Curr. Opin. Biotech. 15 (2004) 487. J.A. Queiroz, C.T. Tomaz, J.M.S. Cabral, J. Biotech. 87 (2001) 143. S.N. Timasheff, H. Inoue, Biochemistry 7 (1968) 2501. W.A.C.H. Melander, Arch. Biochem. Biophys. 183 (1977) 200. P. Strop, F. Mikes, Z. Chytilova, J. Chromatogr. 156 (1978) 239. J.L. Fausnaugh, F.E. Regnier, J. Chromatogr. 359 (1986) 131. C. Machold, K. Deinhofer, R. Hahn, A. Jungbauer, J. Chromatogr. A 972 (2002) 3. R. Hahn, K. Deinhofer, C. Machold, A. Jungbauer, J. Chromatogr. B 790 (2003) 99.

94

E. Haimer et al. / J. Chromatogr. A 1139 (2007) 84–94

[16] F. Xia, D. Nagrath, S.M. Cramer, J. Chromatogr. A 1079 (2005) 229. [17] A. Mahn, J.A. Asenjo, Biotech. Adv. 23 (2005) 359. [18] A. Mahn, G. Zapata-Torres, J.A. Asenjo, J. Chromatogr. A 1066 (2005) 81. [19] J.C. Salgado, I. Rapaport, J.A. Asenjo, J. Chromatogr. A 1107 (2006) 110. [20] T.T. Jones, E.J. Fernandez, J. Colloid Inter. Sci. 259 (2003) 27. [21] J.L. McNay, E.J. Fernandez, J. Chromatogr. A 849 (1999) 135. [22] J.L.M. McNay, E.J. Fernandez, Biotech. Bioeng. 76 (2001) 224. [23] J.L.M. McNay, J.P. O’Connell, E.J. Fernandez, Biotech. Bioeng. 76 (2001) 233. [24] B.A. Snopok, E.V. Kostyukevich, Anal. Biochem. 348 (2006) 222. [25] A. Jungbauer, C. Machold, R. Hahn, J. Chromatogr. A 1079 (2005) 221. [26] S.L. Wu, A. Figueroa, B.L. Karger, J. Chromatogr. 371 (1986) 3. [27] S.L. Wu, K. Benedek, B.L. Karger, J. Chromatogr. 359 (1986) 3. [28] S.U. Sane, S.M. Cramer, T.M. Przybycien, J. Chromatogr. A 849 (1999) 149.

[29] M. Hanson, K.K. Unger, R. Denoyel, J. Rouquerol, J. Biochem. Biophys. Methods 29 (1994) 283. [30] I. Lundstrom, Progress in Colloid and Polymer Science, 1985. p. 76. [31] H. Yang, M.R. Etzel, Ind. Eng. Chem. Res. 42 (2003) 890. [32] A. Tscheliessnig, R. Hahn, A. Jungbauer, J. Chromatogr. A 1069 (2005) 23. [33] W. Hao, J. Wang, J. Chromatogr. A 1063 (2005) 47. [34] A. Cavazzini, F. Dondi, A. Jaulmes, C. Vidal-Madjar, A. Felinger, Anal. Chem. 74 (2002) 6269. [35] G. Golshan-Shirazi, J. Chromatogr. 603 (1992) 1. [36] H.P. Jennissen, Methods Mol. Biol. 305 (2005) 81. [37] H.P. Jennissen, A. Demiroglou, J. Chromatogr. A 1109 (2006) 197. [38] H.P. Jennissen, J. Chromatogr. 159 (1978) 71. [39] J. Mollmann, Frokjaer, Eur. J. Pharm. Sci. 27 (2006) 194. [40] J. Marsh, Sferrazza, Colloid Surf. B 23 (2002) 31. [41] Welzel, Thermochim. Acta 382 (2002) 175. [42] C. Norde, J. Biotech. 79 (2000) 259.