Evaluation of intrinsic immobilized kinetics in hollow fiber reactor systems Steven R. Reiken, Richard J. Knob and Daina M. Briedis Department of Chemical Engineering, Michigan State University, East Lansing, MI
Immobilized cell and enzyme hollow fiber reactors have been developed.fOr c~ variety ~f" biochemical and biomedical applications. Reported mathematical models .for predicting substrate conversion in these reactors have been limited in accuracy because ~)]"the use o f free-solution kinetic parameters. This paper describes a method fi~r determining the intrinsic kinetics o f enzymes immobilized in hollow fiber reactor systems using a mathematical model.for dif'J}~sion and reaction in porous media and an optimization procedure to fit intrinsic kinetic parameters to experimental data. Two enzymes, a thermophilic [3-galactosidase that exhibits product inhibition and t.-lysine e~-oxidase, were used in the analysis. The intrinsic kinetic" parameters show that immobilization enhanced the activity o f the [3galactosidase while decreasing the activity ~[" L-lysine c~-oxidase. Both immobilized enzymes had higher K~n values than did the soluble enzyme, indicating less aJfinity fi~r the substrate. These results are used to illustrate the significant improvement in the ability to predict substrate conversion in hollow fiber reactors.
Keywords: Immobilized enzymes; hollow fiber reactor, intrinsic enzyme kinetics; modeling
Introduction As shown in F i g u r e 1, a hollow fiber is physically divided into three regions: the lumen (region I), the ultrathin membrane (region 2), and the spongy layer (region 3). Over the past decade, development of these fibers has stimulated interest in designing efficient catalytic reactors by immobilizing enzymes (or cells) within the membrane spongy layer. A single-fiber reactor (SFR) shown in F i g u r e 2 was used in developmental work described here because of its convenience, simplicity, and small amounts of biochemicals needed for testing. In normal SFR operation, a laminar feed stream in the lumen is a continuous source of substrate molecules that transfer via diffusion and, under special operating conditions, convection through the ultrathin m e m b r a n e into the annular spongy layer where the enzymatic reaction occurs. The two e n z y m e systems that we have studied are the L-lysine c~-oxidase/catalase and the /3-galactosidase immobilized e n z y m e reactors. These reactors have important practical applications. The L-lysine c~-
Address reprint requests to Dr. Briedis at the Department of Chemical Engineering, Michigan State University, East Lansing, MI, 48824-1226, USA Received 5 June 1989; revised 23 October 1989 736
Enzyme Microb. Technol., 1990, vol. 12, October
oxidase/catalase coimmobilized reactor is currently being investigated as a clinical tool in the treatment of leukemia.I Immobilized thermophilic/3-galactosidase, which hydrolyses the milk sugar lactose, has been studied for use in the dairy industry as a means of improving digestibility and marketability of dairy products and in the treatment of dairy waste streams. 2 Several mathematical models have been developed to predict substrate conversions in enzymatic hollow fiber reactors. 3-6 The most complete models consider mass transfer in the three fiber regions, axial flow and radial diffusion in the lumen, radial diffusion across the ultrathin membrane, and radial diffusion and subsequent reaction in the spongy layer. Other mass transport mechanisms, including axial diffusion and bulk flow across the membrane due to transmembrane pressure gradients, are assumed negligible. One limitation of these earlier models is that the effect of immobilization on the intrinsic e n z y m e kinetics is neglected by assuming that the reaction rate follows free-solution kinetics. The justification typically given for this assumption is that the spongy layer is 80-90% void, allowing the immobilized e n z y m e to remain in a freesolution-like environment; we show this assumption to be incorrect. In this paper, we describe the development and application o f a model to determine the intrinsic kinetics of enzymes immobilized in hollow fiber reactor systems. © 1990 Butterworth-Heinemann
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al.
Lumen_~, A ~ [ ~ k
jW
porous sponge
%--i
Membrane ~
Porous S p o ~
~
....."~............ "~....... "~'~] memb
~]
Theory L-Lysine a-oxidase obeys a Michaelis-Menten kinetic relationship with no inhibition (from catalase or any of the reaction products). The kinetics of/3-galactosidase are more complex in that galactose, a product of the reaction, is a competitive inhibitor of the enzyme.
L-Lysine a-oxidase Figure 1 The cylindrical geometry and dimensions of the hollow fiber (figure not drawn to scale)
Background One of the more important concerns in the modeling of a hollow fiber reactor system is the effect that the immobilization has on the intrinsic kinetics of the enzymes. This ultimately influences overall reactor design and performance. In most heterogeneous catalysis systems, conversion is viewed as a competition between the rate of reaction and the rate of transport. This has been modeled for a variety of enzymatic reaction schemes by Moo-Young and Kobayashi. 7 The model developed is expressed in terms of an effectiveness factor (E) and a generalized modulus (m). The effectiveness factor and modulus used by these authors are defined below: E = experimentally observed reaction rate ideal free-solution reaction rate
(1)
Since the SFR is operated at total recycle, it can be modeled as a batch reactor. For L-lysine a-oxidase coimmobilized with catalase, the rate of the reaction in the hollow fiber recycle reactor may be represented by: Rate of reaction -
dS VmS - E . - dt S + Km
(3)
In this equation, S is the substrate concentration in the lumen; E is the effectiveness factor that is determined using the Moo-Young and Kobayashi model for the case with no inhibition; K,, is the immobilized intrinsic Michaelis constant; and Vm is an adjusted maximum reaction rate that is equal to Vmax(/zmol of substrate reacted per minute) divided by the volume of the catalyst (V,,). This adjustment is made to make equation 3 dimensionally correct. It is seen that the kinetic parameters affect the rate of reaction in the SFR dir e c t l y - t h r o u g h the right-hand side of equation 3 - - a s well as indirectly through the kinetic and transport parameters in the effectiveness factor.
~-Galactosidase m - ~
Ds r(S) dS
(2)
where s is the bulk concentration of substrate in the lumen; L is the characteristic length and is defined as the ratio of catalyst volume (Vc) to lumen surface area (AL)8; r(S) is the intrinsic reaction rate, which should represent the immobilized enzyme kinetics; and D~ is the in situ substrate diffusivity. The modulus may be viewed as a ratio of substrate reaction rate to substrate diffusion rate. Moo-Young and Kobayashi have developed a simplified equation for the effectiveness factor in a hollow fiber membrane that avoids the rigorous integration and solution of a more complex set of equations. 7 In this derivation, the enzyme is assumed to be evenly distributed within the reactor with negligible changes in substrate and product diffusivities; the immobilized enzymes are assumed to obey Michaelis-Menten kinetics (with and without inhibition, depending on the enzyme being studied). The equations resulting from the Moo-Young and Kobayashi analysis show the effectiveness factor for the reactor to be dependent on the intrinsic kinetic parameters of the immobilized enzyme. As will be described in the following section, this model was used to obtain intrinsic kinetics for two enzymes with distinctly different kinetics: L-lysine a-oxidase and/3-galactosidase.
The rate of the reaction for the immobilized/3-galactosidase reactor is similar to equation 3 with the addition of the inhibition term: Rate of reaction -
dS Vm S - E. (4) dt S + Kin(1 + P/Kg)
where P and Ki are the concentration of product (inhibitor) and the competitive inhibition constant, respectively. The effectiveness factor is found from the model for product inhibition. 7 D~ and D o represent diffusivities of substrate (lactose) and product (glucose or
Shl'side__~.~...~,/
H
F
.........~ .............4
c iI.
P"
L ~
I-,,
V/ Tube-side
./
'~
(lumen of HF)
Figure 2 Single-fiber reactor (SFR). Shell material is borosilicate glass, 21.5 cm overall length, 0.8 cm o.d., tapered to 0.5 cm diameter x 1 cm length, end tubes with side tubes 0.5 cm dia. × 1.5 cm length; fittings illustrated on left were applied to both ends of the reactor: L = male and female Luer lock fittings; C = Tygon tube; HF = hollow ultrafiltration fiber. The hollow fiber was retained by a plug of epoxy potting resin (Dew Chemical) between the Luer fittings and the hollow fiber
Enzyme Microb. Technol., 1990, vol. 12, October
737
Papers galactose) within the fiber. Lumen substrate concentration (S) and the initial substrate concentration (So) can be used to calculate the product concentration (P = So - S). Typically, most investigators have calculated effectiveness factors and moduli based on free-solution enzyme kinetics. Strictly speaking, the effectiveness factor should be based on the maximum possible reaction rate in situ, i.e., the intrinsic kinetics of the enzyme in the immobilized state. The above equations may be used in data analysis in order to obtain these intrinsic parameters. Since the effectiveness factor depends on the diffusivities of the substrate and product within the hollow fiber membrane (through the generalized modulus), the first step in our approach was to determine these diffusivities independently.
permeabilities and diffusivities of the transported species may then be calculated directly.
Materials and methods Materials All ultrafiltration (UF) fibers were donated by Romicon, Inc. (Woburn, MA). Lactose, galactose, glucose, o-lysine, L-lysine, ~x-ketoglutarate, NADH, saccharopine dehydrogenase, trichloroacetic acid, and catalase were purchased from Sigma Chemical Co. (St. Louis, MO). The L-lysine a-oxidase was donated by Yamasa Shoyu Co., Ltd. (Tokyo, Japan), and the/~-galactosidase was donated by Dr. Mansel Griffiths of the Hannah Research Institute (Ayr, Scotland). Folin phenol reagent for the Lowry protein assay was manufactured by EM Industries, Inc. (Gibbstown, N J).
Permeability and diffusivity determination In most permeability studies involving hollow fiber membranes, the overall permeability has been determined by following the concentration changes between the tube (lumen)- and shell-side of the hollow fiber in a dialysis mode of operation. 9-~j Substrate solution is pumped through the tube-side, the substrate diffuses across the fiber wall, and is collected in a " b l a n k " solution flowing countercurrently on the shell-side. Material balances between the tube-side and the shell-side are used to calculate the overall membrane permeability (K0) for the transported species (see equation 5 below). The diffusion coefficients may be calculated from the overall membrane permeability as follows. Using the wet membrane thickness as the diffusion path length (0.0445 cm), the effective diffusivities are calculated as Deft- = K0 × wet membrane thickness. 9 Since glucose and galactose have the same molecular weight, their diffusivities are assumed to be equal. Similarly, the diffusivity of L-lysine was assumed to be equal to that of D-[ysine. D-Lysine was used as the diffusing species so that the effect of the presence of immobilized enzyme on substrate diffusivity could be evaluated without interference from the enzymatic reaction. The overall membrane permeabilities for lactose, galactose, and D-lysine for a hollow fiber membrane were calculated with the material balance equation used by Park et al.12: - I n (m - (Vt + Vs) S~] Koa V,(~,0 s,0) / = ~ (v, + v,)t -
(5)
where Vt and V, are the volumes of solute solution in the tube- and shell-sides, respectively; St and Ss are the concentrations of solute in the tube- and shellsides; St0 and Sso are the initial concentrations of solute in the tube- and shell-sides; m = StVt + SsV~ ; A is the mass transfer area of the hollow fiber membrane (7.75 cm2); and t is the time of sample. Since these are all known or measured quantities, the term {-ln[(m (V~ + Vs)Ss)/(Vt(Sto - Ss0))]} may be plotted versus time. The slope of the plot is equal to KoA(Vt + Vs)/ (VtV,) as shown by equation 5. The overall membrane 738
Enzyme Microb. Technol., 1990, vol. 12, October
Analyses Protein was measured according to Lowry et al.13 Lysine concentrations were determined by the saccharopine dehydrogenase method described elsewhere. TM o-Lysine was assayed by the o-amino acid oxidase technique described by Stokes and Gunness. 15 Glucose determinations were performed by a phosphatase-glucose oxidase (PGO) method (Sigma Diagnostics, Procedure #510) and by HPLC analysis.
Permeability/diffusivity experiments A single-fiber reactor (SFR) was constructed and used in experiments designed to evaluate the permeability of substrate and product through the hollow fiber wall. The SFR consists of a single polyamide hollow fiber with a nominal molecular weight cutoff of 10,000 (PAl0 fiber) encased in a glass shell giving a shell-andtube configuration (Figure 2). These reactors are identical to those discussed by Reiken and Briedis. ~ The SFR used in this study was assumed to be representative of the characteristics of other PAl0 fibers, although the characteristics of individual PAl0 fibers may vary slightly. The SFR was used in a dialysis mode in which the test species was circulated through the tube-side, and a "blank" solution was circulated through the shellside to collect the diffusing species. The dialysis configuration used was similar to the one described by Farreli and Babb, ~° which uses a countercurrent flow pattern through the reactor. Pulse dampeners were placed upstream of the reactor to reduce the effects of pulsatile flow. Lactose and glucose permeability experiments were conducted without the presence of the immobilized/3-galactosidase; the D-lysine experiments were conducted both in the absence and presence of immobilized L-lysine c~-oxidase. One PAl0 SFR was used throughout each series of experiments. Backflush loading, in which an enzyme solution is pumped through the UF fibers in a reversed UF mode, was chosen as the technique for immobilizing the enzymes in the hollow fiber. This was accomplished according to the protocol described elsewhere.I
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al. ml/min. Flow rates were chosen so as to minimize liquid film mass transfer resistances.
Bypass
Reactor operation
/~n ! ~xes
Recycle
Figure 3 Hollow fiber reactor system (HFR). Res = reservoir flask; P = p u m p ; f l , f2 = flowmeters; p l = lumen side inlet port; p2 = shell-side inlet port; p3 = lumen outlet port; p4 = shell-side outlet port; t l = 15 psig pressure transducer; t2 = 5 psi differential pressure transducer; O = outlet sample port; X = tubing clamps for stopping flow
In the D-lysine permeability experiments, the SFR was maintained at a temperature of 37°C in a water bath. The shell- and tube-side solutions were pumped through bypass lines in each independent flow circuit until equilibria in temperature and in flow rate were reached (approximately 10 min). Once overall equilibrium was achieved, the solutions were allowed to pass through the SFR. The hydrostatic pressure difference between each side of the reactor was monitored and minimized by creating back pressure on the low pressure side. This was done to prevent any pressuredriven enzyme leakage and to avoid any convective flows due to transmembrane pressure gradients. Samples were collected over an 8-h period. A similar procedure was used in the lactose and glucose permeability experiments. In these experiments, the SFR was maintained at a temperature of 54.5°C, and a 147 mM lactose solution was pumped through the tube-side loop. Experiments were repeated at several flow rates. The shell-side flow rates ranged from 3.6 to 18.2 ml/min, while the tube-side flow rates were fixed at values between 6.0 and 12.6
Table 1
Backflush loading (described above) was used for enzyme immobilization. The SFR was installed in a laboratory reactor system (Figure 3). The fluid conducting elements of the system consisted of Tygon tubing (A inch i.d.) with polyethylene T- and quick-disconnect connectors at the junctions. All tubing was wrapped with foam insulation. Valves at the junctions were adjusted to determine the circulation pattern. Both the substrate reservoir and the reactor were held at a constant temperature (54.5°C for fl-galactosidase, 37°C for L-lysine a-oxidase) in a water bath. During reactor experiments, the liquid-free shellside of the system was closed. The reservoir contained either a 2.0 mM lysine solution or a 146.7 mM lactose solution. With the recycle loop opened, the solution was pumped from the reservoir through the fiber lumen. To collect samples, the recycle loop was closed and the sample port was opened. This mode of operation represents a total recycle reactor that may be modeled as a batch reactor (equations 3 and 4). Data were collected and analyzed over a 48-h period. An extended-time-period experiment was necessary so that a broad range of substrate concentration was covered and an accurate determination of the intrinsic kinetics could be made. Samples were taken at regular time intervals and were promptly analyzed by the methods described above.
Results
Permeability/diffusivity studies The permeability and diffusivity values for lactose and glucose through the wall of a clean PAl0 hollow fiber without the presence of enzyme are shown in Table 1. Park et al.12 reported a higher permeability for lactose in polysulfone hollow fibers (3.95 x 10-5 cm/min). Despite their lower permeability, we selected polyamide fibers because of their better compatibility with the enzyme compared to polysulfone fibers. ~6 The calculated diffusivities are somewhat lower than are freesolution diffusivities ~7for lactose (8.3 x 10-6 cm2/sec) and for glucose (1.17 × 10-5 cmZ/sec). These results may be due to the mass transfer resistance of the hol-
Permeability and diffusivity values for substrate and products of enzyme reactions D-Lysine
Overall membrane permeability (cm/min x 103 Effective diffusivity De. (cm/s) x 102
No enzyme
With
Lactose
Glucose
present
L-lysine e-oxidase
1.15
2.49
2.45
0.237
0.86
1.85
1.82
0.176
Enzyme Microb. Technol., 1990, vol. 12, October
7:39
z
Papers
40~
50-
Determination o f intrinsic immobil&ed kinetics The data from the extended reactor operation experiments were plotted as time versus /xmol of glucose produced for the/3-galactosidase study (Figure 4) and time versus L-iysine concentration for the oxidase experiments (Figure 5). Time was plotted on the ordinate to simplify the fitting of the data to a reaction rate expression that describes the dependence of substrate or product concentration on time. The following mathematical expressions were found to best fit the data:
20-
lO
/3-galactosidase: o
, ~
0
1000
2000
3000
4000
,
50~00 60'00
7000
Rate =
M i c r o m o l s of Glucose P r o d u c e d Data from a 48-h/~-galactosidase reactor performance
Figure 4 study
30 O
E
20
\
0.0'
' '0'.~'
' 'o'.a'
' '1:2~'
Lysine C o n c e n t r a t i o n
1.6
2.0'
(raM)
Figure 5 Data from a 48-h [-lysine a~-oxidase reactor performance study
low fiber membrane, an effect usually neglected in hollow fiber reactor modeling. Since the SFR is operated with a relatively concentrated enzyme solution in the pores of the fiber wall, the in situ diffusivities should be even lower. This was evaluated for the case of the lysine/lysine a-oxidase system. The permeability of D-lysine through the PAl0 fiber was studied in more detail in that experiments were conducted both with and without L-lysine a-oxidase immobilized within the fiber. Results are shown in Table 1. The diffusivity of amino acids in free-solution has been reported to be on the order of 10 -6 cm2/ sec, 18-2° which corresponds to the value found from the experiments with no enzyme immobilized within the fiber wall. It should be noted that the diffusivity of lysine decreased by an order of magnitude in the presence of immobilized enzyme. This is important because the use of free-solution diffusivities in hollow fiber reactor modeling will lead to errors in calculated values of moduli and effectiveness factors. 740
Enzyme Microb. Technol., 1990, vol. 12, October
(! 7)
Q(1) x Q(2) × exp(Q(2) x P)
L-lysine oPoxidase: Rate =
50-
1
S Q(3) x S + Q(4)
(i 8)
where Q(l), Q(2), Q(3), and Q(4) are the curve-fitting parameters; P represents the micromoles of glucose produced; and S is the lysine concentration. The correlations between the data and the fit expressions were greater than 99%. The optimization program 2~ was written to use the experimentally measured reaction rates to fit the intrinsic immobilized kinetics in the model using the Moo-Young and Kobayashi equations for the effectiveness factor and equation 3 for L-lysine a-oxidase/ catalase or equation 4 for/3-galactosidase. This program uses an initial guess for the immobilized kinetic parameters and evaluates the calculated reaction rate from the model at several concentrations of substrate (as for L-lysine a-oxidase) or product (as for/3-galactosidase). The calculated reaction rates are then compared to those obtained experimentally. The program minimizes the sum of the square of the residuals between the observed and calculated reaction rates by adjusting the values of the kinetic parameters. The results of the optimization are shown in Table 2. Freesolution kinetics are also reported for comparison. Results similar to those listed in Table 2 were obtained using different levels of enzyme loading and Table 2
Enzyme kinetic parameters
Enzyme
Intrinsic (immobilized)
L-Lysine c~-oxidase units/mg
V,~lax = 7.44 unitsa/mg
V,~x
K~n
K,,~ - 0.06 mM
0.688 mM
/~-galactosidase Vma× -- 14.53 unitsb/mg K m = 20.35 mM K , - 0.765 mM
Free-solution 45.6 units/rag
Vmax = 2.08 units/mg K m = 3.69 mM Kj - 7.16 mM
" One unit will consume 1 /~mol of 02 per minute with L-lysine as substrate at pH 7.4 and 30°C b One unit hydrolyses 1/zmol of lactose per minute to e q u i m o l a r glucose and galactose at pH 7.3 and 37°C
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al. 100--
• . o v---
Immobilized cell and enzyme hollow fiber reactors have been developed.fOr c~ variety ~f" biochemical and biomedical applications. Reported mathematical models .for predicting substrate conversion in these reactors have been limited in accuracy because ~)]"the use o f free-solution kinetic parameters. This paper describes a method fi~r determining the intrinsic kinetics o f enzymes immobilized in hollow fiber reactor systems using a mathematical model.for dif'J}~sion and reaction in porous media and an optimization procedure to fit intrinsic kinetic parameters to experimental data. Two enzymes, a thermophilic [3-galactosidase that exhibits product inhibition and t.-lysine e~-oxidase, were used in the analysis. The intrinsic kinetic" parameters show that immobilization enhanced the activity o f the [3galactosidase while decreasing the activity ~[" L-lysine c~-oxidase. Both immobilized enzymes had higher K~n values than did the soluble enzyme, indicating less aJfinity fi~r the substrate. These results are used to illustrate the significant improvement in the ability to predict substrate conversion in hollow fiber reactors.
Keywords: Immobilized enzymes; hollow fiber reactor, intrinsic enzyme kinetics; modeling
Introduction As shown in F i g u r e 1, a hollow fiber is physically divided into three regions: the lumen (region I), the ultrathin membrane (region 2), and the spongy layer (region 3). Over the past decade, development of these fibers has stimulated interest in designing efficient catalytic reactors by immobilizing enzymes (or cells) within the membrane spongy layer. A single-fiber reactor (SFR) shown in F i g u r e 2 was used in developmental work described here because of its convenience, simplicity, and small amounts of biochemicals needed for testing. In normal SFR operation, a laminar feed stream in the lumen is a continuous source of substrate molecules that transfer via diffusion and, under special operating conditions, convection through the ultrathin m e m b r a n e into the annular spongy layer where the enzymatic reaction occurs. The two e n z y m e systems that we have studied are the L-lysine c~-oxidase/catalase and the /3-galactosidase immobilized e n z y m e reactors. These reactors have important practical applications. The L-lysine c~-
Address reprint requests to Dr. Briedis at the Department of Chemical Engineering, Michigan State University, East Lansing, MI, 48824-1226, USA Received 5 June 1989; revised 23 October 1989 736
Enzyme Microb. Technol., 1990, vol. 12, October
oxidase/catalase coimmobilized reactor is currently being investigated as a clinical tool in the treatment of leukemia.I Immobilized thermophilic/3-galactosidase, which hydrolyses the milk sugar lactose, has been studied for use in the dairy industry as a means of improving digestibility and marketability of dairy products and in the treatment of dairy waste streams. 2 Several mathematical models have been developed to predict substrate conversions in enzymatic hollow fiber reactors. 3-6 The most complete models consider mass transfer in the three fiber regions, axial flow and radial diffusion in the lumen, radial diffusion across the ultrathin membrane, and radial diffusion and subsequent reaction in the spongy layer. Other mass transport mechanisms, including axial diffusion and bulk flow across the membrane due to transmembrane pressure gradients, are assumed negligible. One limitation of these earlier models is that the effect of immobilization on the intrinsic e n z y m e kinetics is neglected by assuming that the reaction rate follows free-solution kinetics. The justification typically given for this assumption is that the spongy layer is 80-90% void, allowing the immobilized e n z y m e to remain in a freesolution-like environment; we show this assumption to be incorrect. In this paper, we describe the development and application o f a model to determine the intrinsic kinetics of enzymes immobilized in hollow fiber reactor systems. © 1990 Butterworth-Heinemann
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al.
Lumen_~, A ~ [ ~ k
jW
porous sponge
%--i
Membrane ~
Porous S p o ~
~
....."~............ "~....... "~'~] memb
~]
Theory L-Lysine a-oxidase obeys a Michaelis-Menten kinetic relationship with no inhibition (from catalase or any of the reaction products). The kinetics of/3-galactosidase are more complex in that galactose, a product of the reaction, is a competitive inhibitor of the enzyme.
L-Lysine a-oxidase Figure 1 The cylindrical geometry and dimensions of the hollow fiber (figure not drawn to scale)
Background One of the more important concerns in the modeling of a hollow fiber reactor system is the effect that the immobilization has on the intrinsic kinetics of the enzymes. This ultimately influences overall reactor design and performance. In most heterogeneous catalysis systems, conversion is viewed as a competition between the rate of reaction and the rate of transport. This has been modeled for a variety of enzymatic reaction schemes by Moo-Young and Kobayashi. 7 The model developed is expressed in terms of an effectiveness factor (E) and a generalized modulus (m). The effectiveness factor and modulus used by these authors are defined below: E = experimentally observed reaction rate ideal free-solution reaction rate
(1)
Since the SFR is operated at total recycle, it can be modeled as a batch reactor. For L-lysine a-oxidase coimmobilized with catalase, the rate of the reaction in the hollow fiber recycle reactor may be represented by: Rate of reaction -
dS VmS - E . - dt S + Km
(3)
In this equation, S is the substrate concentration in the lumen; E is the effectiveness factor that is determined using the Moo-Young and Kobayashi model for the case with no inhibition; K,, is the immobilized intrinsic Michaelis constant; and Vm is an adjusted maximum reaction rate that is equal to Vmax(/zmol of substrate reacted per minute) divided by the volume of the catalyst (V,,). This adjustment is made to make equation 3 dimensionally correct. It is seen that the kinetic parameters affect the rate of reaction in the SFR dir e c t l y - t h r o u g h the right-hand side of equation 3 - - a s well as indirectly through the kinetic and transport parameters in the effectiveness factor.
~-Galactosidase m - ~
Ds r(S) dS
(2)
where s is the bulk concentration of substrate in the lumen; L is the characteristic length and is defined as the ratio of catalyst volume (Vc) to lumen surface area (AL)8; r(S) is the intrinsic reaction rate, which should represent the immobilized enzyme kinetics; and D~ is the in situ substrate diffusivity. The modulus may be viewed as a ratio of substrate reaction rate to substrate diffusion rate. Moo-Young and Kobayashi have developed a simplified equation for the effectiveness factor in a hollow fiber membrane that avoids the rigorous integration and solution of a more complex set of equations. 7 In this derivation, the enzyme is assumed to be evenly distributed within the reactor with negligible changes in substrate and product diffusivities; the immobilized enzymes are assumed to obey Michaelis-Menten kinetics (with and without inhibition, depending on the enzyme being studied). The equations resulting from the Moo-Young and Kobayashi analysis show the effectiveness factor for the reactor to be dependent on the intrinsic kinetic parameters of the immobilized enzyme. As will be described in the following section, this model was used to obtain intrinsic kinetics for two enzymes with distinctly different kinetics: L-lysine a-oxidase and/3-galactosidase.
The rate of the reaction for the immobilized/3-galactosidase reactor is similar to equation 3 with the addition of the inhibition term: Rate of reaction -
dS Vm S - E. (4) dt S + Kin(1 + P/Kg)
where P and Ki are the concentration of product (inhibitor) and the competitive inhibition constant, respectively. The effectiveness factor is found from the model for product inhibition. 7 D~ and D o represent diffusivities of substrate (lactose) and product (glucose or
Shl'side__~.~...~,/
H
F
.........~ .............4
c iI.
P"
L ~
I-,,
V/ Tube-side
./
'~
(lumen of HF)
Figure 2 Single-fiber reactor (SFR). Shell material is borosilicate glass, 21.5 cm overall length, 0.8 cm o.d., tapered to 0.5 cm diameter x 1 cm length, end tubes with side tubes 0.5 cm dia. × 1.5 cm length; fittings illustrated on left were applied to both ends of the reactor: L = male and female Luer lock fittings; C = Tygon tube; HF = hollow ultrafiltration fiber. The hollow fiber was retained by a plug of epoxy potting resin (Dew Chemical) between the Luer fittings and the hollow fiber
Enzyme Microb. Technol., 1990, vol. 12, October
737
Papers galactose) within the fiber. Lumen substrate concentration (S) and the initial substrate concentration (So) can be used to calculate the product concentration (P = So - S). Typically, most investigators have calculated effectiveness factors and moduli based on free-solution enzyme kinetics. Strictly speaking, the effectiveness factor should be based on the maximum possible reaction rate in situ, i.e., the intrinsic kinetics of the enzyme in the immobilized state. The above equations may be used in data analysis in order to obtain these intrinsic parameters. Since the effectiveness factor depends on the diffusivities of the substrate and product within the hollow fiber membrane (through the generalized modulus), the first step in our approach was to determine these diffusivities independently.
permeabilities and diffusivities of the transported species may then be calculated directly.
Materials and methods Materials All ultrafiltration (UF) fibers were donated by Romicon, Inc. (Woburn, MA). Lactose, galactose, glucose, o-lysine, L-lysine, ~x-ketoglutarate, NADH, saccharopine dehydrogenase, trichloroacetic acid, and catalase were purchased from Sigma Chemical Co. (St. Louis, MO). The L-lysine a-oxidase was donated by Yamasa Shoyu Co., Ltd. (Tokyo, Japan), and the/~-galactosidase was donated by Dr. Mansel Griffiths of the Hannah Research Institute (Ayr, Scotland). Folin phenol reagent for the Lowry protein assay was manufactured by EM Industries, Inc. (Gibbstown, N J).
Permeability and diffusivity determination In most permeability studies involving hollow fiber membranes, the overall permeability has been determined by following the concentration changes between the tube (lumen)- and shell-side of the hollow fiber in a dialysis mode of operation. 9-~j Substrate solution is pumped through the tube-side, the substrate diffuses across the fiber wall, and is collected in a " b l a n k " solution flowing countercurrently on the shell-side. Material balances between the tube-side and the shell-side are used to calculate the overall membrane permeability (K0) for the transported species (see equation 5 below). The diffusion coefficients may be calculated from the overall membrane permeability as follows. Using the wet membrane thickness as the diffusion path length (0.0445 cm), the effective diffusivities are calculated as Deft- = K0 × wet membrane thickness. 9 Since glucose and galactose have the same molecular weight, their diffusivities are assumed to be equal. Similarly, the diffusivity of L-lysine was assumed to be equal to that of D-[ysine. D-Lysine was used as the diffusing species so that the effect of the presence of immobilized enzyme on substrate diffusivity could be evaluated without interference from the enzymatic reaction. The overall membrane permeabilities for lactose, galactose, and D-lysine for a hollow fiber membrane were calculated with the material balance equation used by Park et al.12: - I n (m - (Vt + Vs) S~] Koa V,(~,0 s,0) / = ~ (v, + v,)t -
(5)
where Vt and V, are the volumes of solute solution in the tube- and shell-sides, respectively; St and Ss are the concentrations of solute in the tube- and shellsides; St0 and Sso are the initial concentrations of solute in the tube- and shell-sides; m = StVt + SsV~ ; A is the mass transfer area of the hollow fiber membrane (7.75 cm2); and t is the time of sample. Since these are all known or measured quantities, the term {-ln[(m (V~ + Vs)Ss)/(Vt(Sto - Ss0))]} may be plotted versus time. The slope of the plot is equal to KoA(Vt + Vs)/ (VtV,) as shown by equation 5. The overall membrane 738
Enzyme Microb. Technol., 1990, vol. 12, October
Analyses Protein was measured according to Lowry et al.13 Lysine concentrations were determined by the saccharopine dehydrogenase method described elsewhere. TM o-Lysine was assayed by the o-amino acid oxidase technique described by Stokes and Gunness. 15 Glucose determinations were performed by a phosphatase-glucose oxidase (PGO) method (Sigma Diagnostics, Procedure #510) and by HPLC analysis.
Permeability/diffusivity experiments A single-fiber reactor (SFR) was constructed and used in experiments designed to evaluate the permeability of substrate and product through the hollow fiber wall. The SFR consists of a single polyamide hollow fiber with a nominal molecular weight cutoff of 10,000 (PAl0 fiber) encased in a glass shell giving a shell-andtube configuration (Figure 2). These reactors are identical to those discussed by Reiken and Briedis. ~ The SFR used in this study was assumed to be representative of the characteristics of other PAl0 fibers, although the characteristics of individual PAl0 fibers may vary slightly. The SFR was used in a dialysis mode in which the test species was circulated through the tube-side, and a "blank" solution was circulated through the shellside to collect the diffusing species. The dialysis configuration used was similar to the one described by Farreli and Babb, ~° which uses a countercurrent flow pattern through the reactor. Pulse dampeners were placed upstream of the reactor to reduce the effects of pulsatile flow. Lactose and glucose permeability experiments were conducted without the presence of the immobilized/3-galactosidase; the D-lysine experiments were conducted both in the absence and presence of immobilized L-lysine c~-oxidase. One PAl0 SFR was used throughout each series of experiments. Backflush loading, in which an enzyme solution is pumped through the UF fibers in a reversed UF mode, was chosen as the technique for immobilizing the enzymes in the hollow fiber. This was accomplished according to the protocol described elsewhere.I
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al. ml/min. Flow rates were chosen so as to minimize liquid film mass transfer resistances.
Bypass
Reactor operation
/~n ! ~xes
Recycle
Figure 3 Hollow fiber reactor system (HFR). Res = reservoir flask; P = p u m p ; f l , f2 = flowmeters; p l = lumen side inlet port; p2 = shell-side inlet port; p3 = lumen outlet port; p4 = shell-side outlet port; t l = 15 psig pressure transducer; t2 = 5 psi differential pressure transducer; O = outlet sample port; X = tubing clamps for stopping flow
In the D-lysine permeability experiments, the SFR was maintained at a temperature of 37°C in a water bath. The shell- and tube-side solutions were pumped through bypass lines in each independent flow circuit until equilibria in temperature and in flow rate were reached (approximately 10 min). Once overall equilibrium was achieved, the solutions were allowed to pass through the SFR. The hydrostatic pressure difference between each side of the reactor was monitored and minimized by creating back pressure on the low pressure side. This was done to prevent any pressuredriven enzyme leakage and to avoid any convective flows due to transmembrane pressure gradients. Samples were collected over an 8-h period. A similar procedure was used in the lactose and glucose permeability experiments. In these experiments, the SFR was maintained at a temperature of 54.5°C, and a 147 mM lactose solution was pumped through the tube-side loop. Experiments were repeated at several flow rates. The shell-side flow rates ranged from 3.6 to 18.2 ml/min, while the tube-side flow rates were fixed at values between 6.0 and 12.6
Table 1
Backflush loading (described above) was used for enzyme immobilization. The SFR was installed in a laboratory reactor system (Figure 3). The fluid conducting elements of the system consisted of Tygon tubing (A inch i.d.) with polyethylene T- and quick-disconnect connectors at the junctions. All tubing was wrapped with foam insulation. Valves at the junctions were adjusted to determine the circulation pattern. Both the substrate reservoir and the reactor were held at a constant temperature (54.5°C for fl-galactosidase, 37°C for L-lysine a-oxidase) in a water bath. During reactor experiments, the liquid-free shellside of the system was closed. The reservoir contained either a 2.0 mM lysine solution or a 146.7 mM lactose solution. With the recycle loop opened, the solution was pumped from the reservoir through the fiber lumen. To collect samples, the recycle loop was closed and the sample port was opened. This mode of operation represents a total recycle reactor that may be modeled as a batch reactor (equations 3 and 4). Data were collected and analyzed over a 48-h period. An extended-time-period experiment was necessary so that a broad range of substrate concentration was covered and an accurate determination of the intrinsic kinetics could be made. Samples were taken at regular time intervals and were promptly analyzed by the methods described above.
Results
Permeability/diffusivity studies The permeability and diffusivity values for lactose and glucose through the wall of a clean PAl0 hollow fiber without the presence of enzyme are shown in Table 1. Park et al.12 reported a higher permeability for lactose in polysulfone hollow fibers (3.95 x 10-5 cm/min). Despite their lower permeability, we selected polyamide fibers because of their better compatibility with the enzyme compared to polysulfone fibers. ~6 The calculated diffusivities are somewhat lower than are freesolution diffusivities ~7for lactose (8.3 x 10-6 cm2/sec) and for glucose (1.17 × 10-5 cmZ/sec). These results may be due to the mass transfer resistance of the hol-
Permeability and diffusivity values for substrate and products of enzyme reactions D-Lysine
Overall membrane permeability (cm/min x 103 Effective diffusivity De. (cm/s) x 102
No enzyme
With
Lactose
Glucose
present
L-lysine e-oxidase
1.15
2.49
2.45
0.237
0.86
1.85
1.82
0.176
Enzyme Microb. Technol., 1990, vol. 12, October
7:39
z
Papers
40~
50-
Determination o f intrinsic immobil&ed kinetics The data from the extended reactor operation experiments were plotted as time versus /xmol of glucose produced for the/3-galactosidase study (Figure 4) and time versus L-iysine concentration for the oxidase experiments (Figure 5). Time was plotted on the ordinate to simplify the fitting of the data to a reaction rate expression that describes the dependence of substrate or product concentration on time. The following mathematical expressions were found to best fit the data:
20-
lO
/3-galactosidase: o
, ~
0
1000
2000
3000
4000
,
50~00 60'00
7000
Rate =
M i c r o m o l s of Glucose P r o d u c e d Data from a 48-h/~-galactosidase reactor performance
Figure 4 study
30 O
E
20
\
0.0'
' '0'.~'
' 'o'.a'
' '1:2~'
Lysine C o n c e n t r a t i o n
1.6
2.0'
(raM)
Figure 5 Data from a 48-h [-lysine a~-oxidase reactor performance study
low fiber membrane, an effect usually neglected in hollow fiber reactor modeling. Since the SFR is operated with a relatively concentrated enzyme solution in the pores of the fiber wall, the in situ diffusivities should be even lower. This was evaluated for the case of the lysine/lysine a-oxidase system. The permeability of D-lysine through the PAl0 fiber was studied in more detail in that experiments were conducted both with and without L-lysine a-oxidase immobilized within the fiber. Results are shown in Table 1. The diffusivity of amino acids in free-solution has been reported to be on the order of 10 -6 cm2/ sec, 18-2° which corresponds to the value found from the experiments with no enzyme immobilized within the fiber wall. It should be noted that the diffusivity of lysine decreased by an order of magnitude in the presence of immobilized enzyme. This is important because the use of free-solution diffusivities in hollow fiber reactor modeling will lead to errors in calculated values of moduli and effectiveness factors. 740
Enzyme Microb. Technol., 1990, vol. 12, October
(! 7)
Q(1) x Q(2) × exp(Q(2) x P)
L-lysine oPoxidase: Rate =
50-
1
S Q(3) x S + Q(4)
(i 8)
where Q(l), Q(2), Q(3), and Q(4) are the curve-fitting parameters; P represents the micromoles of glucose produced; and S is the lysine concentration. The correlations between the data and the fit expressions were greater than 99%. The optimization program 2~ was written to use the experimentally measured reaction rates to fit the intrinsic immobilized kinetics in the model using the Moo-Young and Kobayashi equations for the effectiveness factor and equation 3 for L-lysine a-oxidase/ catalase or equation 4 for/3-galactosidase. This program uses an initial guess for the immobilized kinetic parameters and evaluates the calculated reaction rate from the model at several concentrations of substrate (as for L-lysine a-oxidase) or product (as for/3-galactosidase). The calculated reaction rates are then compared to those obtained experimentally. The program minimizes the sum of the square of the residuals between the observed and calculated reaction rates by adjusting the values of the kinetic parameters. The results of the optimization are shown in Table 2. Freesolution kinetics are also reported for comparison. Results similar to those listed in Table 2 were obtained using different levels of enzyme loading and Table 2
Enzyme kinetic parameters
Enzyme
Intrinsic (immobilized)
L-Lysine c~-oxidase units/mg
V,~lax = 7.44 unitsa/mg
V,~x
K~n
K,,~ - 0.06 mM
0.688 mM
/~-galactosidase Vma× -- 14.53 unitsb/mg K m = 20.35 mM K , - 0.765 mM
Free-solution 45.6 units/rag
Vmax = 2.08 units/mg K m = 3.69 mM Kj - 7.16 mM
" One unit will consume 1 /~mol of 02 per minute with L-lysine as substrate at pH 7.4 and 30°C b One unit hydrolyses 1/zmol of lactose per minute to e q u i m o l a r glucose and galactose at pH 7.3 and 37°C
Immobilized kinetics in hollow fiber reactors: S. R. Reiken et al. 100--
• . o v---
