Hydrolysis of Urea by Gelatin-Immobilized Urease: Separation of Kinetic and Diffusion Phenomena in a Model Immobilized-Enzyme Reactor System JOHN PHILIP BOLLMEIER and STANLEY MIDDLEMAN, Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
Summary Experiments and appropriate mathematical models are presented in an attempt to elucidate and separate the effects of mass transfer and immobilization on the apparent kinetics of hydrolysis of urea by urease immobilized within a crosslinked gelatin film. Diffusion of urea through the gelatin matrix appears to exert the major influence on the observed kinetics. Diffusion coefficients are measured, and a model for the “effectiveness factor” is presented, accounting for this aspect of mass transfer control. A secondary, but significant, influence on apparent kinetics arises because the reaction products lead to an increased pH level which, because of diffusion resistance, remains high within the gelatin matrix. For pH levels in the 6.7 to 9.0 range the activity of urease is a strongly decreasing function of pH. An approximate model accounting for ionic equilibrium allows this pH-diffusion effect to be introduced in such a way as to lead to predictions of the apparent kinetics that are compared with experimental observations. Examination of these results indicates that the immobilization procedure leads to some loss of activity due to an interaction of the gelatin crosslinking reaction with the enzyme itself.
INTRODUCTION
The potential of immobilized enzymes as catalysts in chemical reactors has motivated a considerable amount of research activity, and has led to the appearance of a significant amount of literature in this While the variety of enzymes, of carriers for immobilization, and suggested commercial applications confound the field somewhat, certain general features appear that serve as a link from one study to another: 1) Immobilization generally alters the activity of an enzyme relative to its free solution activity. 2) While apparent activity loss may occur due to mass transfer Biotechnology and Bioengineering, Vol. XXI, Pp. 2303-2321 (1979) 0006-3592/79/0021-2303$0 I .OO @ 1979 John Wiley & Sons, Inc.
2304
BOLLMEIER AND MIDDLEMAN
resistance to (or within) the solid carrier to (or within) which the enzyme is immobilized, infrinsic activity changes may occur as well. Changes in intrinsic activity (which we define to be those not explicable as simple effects of mass transfer resistance) may themselves arise from a variety of causes. For example, if immobilization is by covalent bonding to a carrier, the immobilization reaction may inactivate a certain fraction of the enzyme molecules. It is also possible that upon immobilization the immediate surroundings (the microenvironment) of the enzyme may alter their configuration sufficiently to alter activity. This latter effect may occur whether the immobilization is through covalent bonding, or more simply through physical entrapment within the carrier matrix. There is yet another means of activity alteration, but one which in a sense is a combination of mass transfer and microenvironmental effects. If the reaction product has the capacity to alter the reaction rate (two examples would be product inhibition and pH dependence of activity), then immobilization can produce an environment surrounding the enzyme which, because of mass transfer resistance, has a composition different from the surrounding solution. Thus, the enzyme would exhibit a different activity than in free solution. If one wishes to design an immobilized-enzyme reactor, it is necessary to be able to separate these effects on activity, and to determine the conditions under which each of these effects will occur and the extent to which the apparent activity is altered. This paper reports a study of urease immobilized within crosslinked gelatin. Through a combination of experiments and modeling we have been able to determine the separate contributions of the phenomena described above. KINETICS IN FREE SOLUTION
Enzymatic reactions are often observed to obey the MichaelisMenten model, in which the reaction rate is given by
r
=
V,S/(K,
+ S)
Here, S refers to the concentration of reactant (called “substrate” in the biochemical literature). For small concentrations of substrate the kinetics appear to be first order. For large S the kinetics approach zeroth order, and V , is the “maximum velocity” of the reaction. V,, and thus r , are based on unit mass of enzyme, so r is the conversion rate per unit mass of enzyme.
HYDROLYSIS OF UREA
2305
The Michaelis-Menten parameters were measured in our studies from initial rates of reaction carried out at constant temperature in a well-stirred batch reactor. At a given pH value (5 5 pH 5 9) a set of urea solutions was prepared by adding from 0.1 to 0.9 g urea to 100 ml buffer [O.lM TRIS (hydroxymethyl) amino methane]. Each urea solution was then diluted in 200 ml O.1M TRIS with 1mM EDTA, adjusted to the desired pH with sulfuric acid. This solution was in a stirred beaker in a thermostated bath. To this solution, 2 ml 0.1 M TRIS with 1mM EDTA containing 11.7 mg urease (Sigma type 111, lot No. 66c-7550) were added. One ml samples were withdrawn at 1 min intervals for 11 min, and the samples were analyzed for urea by the Levine3 method, using a Leitz model M photometer. Temperature was controlled at 25°C. pH was measured continuously and maintained constant by dropwise addition of 1M H,SO, when necessary. For each experiment the initial rate was determined from the slope of urea concentration versus time data. The K , and V , parameters were then determined by fitting the Michaelis- Menten model to the data. Figures 1 and 2 show the results. A maximum in V , at pH = 6.75 is seen, which is consistent with published results for ~ r e a s e . The ~ , ~ K , values were subject to considerable uncertainty for pH 6 to 7, and it is not clear if a true maximum exists, as the data suggest.
0 4
5
6
?
a
9
PH
Fig. 1. 25°C.
Maximum rate velocity as a function of pH for urease in free solution, at
2306
BOLLMEIER A N D MIDDLEMAN
? 01 5
0 0
I 6
1
I
7
8
a +
9
PH
Fig. 2. Michaelis-Menten constant, K , , as a function of pH for urease in free solution, at 25°C.
OBSERVED IMMOBILIZED UREASE KINETICS
Urease-gelatin membranes were formed by first dissolving 0.8 g pigskin gelatin (Swift’s lot No. 3884-2) in 10 ml O.1M TRIS with ImM EDTA at 35°C. To this solution 70 mg urease were added, and after complete dissolution the solution was poured onto a stretched polyethylene film to form a puddle about 8 cm in diam. Once gelled, the membrane was placed in a forced-air oven and allowed to dry overnight at 27°C. The dried membrane thickness was measured with a micrometer, and then the membrane was swollen by placing it in O.1M TRIS with ImM EDTA, at the desired pH, for 15 min at 5°C. The swollen membrane was then crosslinked in a 10% solution of glutaraldehyde for 15 or 30 sec, depending on the desired degree of crosslinking. The freshly crosslinked membrane was washed well with deionized distilled water before use in the reactor. Figure 3 shows the reactor that was fabricated of acrylic pipe and sheet and had a volume of approximately 70 ml. For the kinetic studies the reactor was operated as a continuous-flow stirred tank at 25°C. The membrane was cut to a 67 mm diam and backed by an identical piece of polyethylene for support. The area of membrane exposed to reactant was 15.55 cm2. Reactor feed was from a stain-
HYDROLYSIS OF UREA
2307
less-steel pressure vessel. The feed stock was a urea solution in 0.1M TRIS with 1mM EDTA. The feed rate was 14 ml/min. Individual membranes varied in wet thickness, but were generaily from 0.5 to 1 mm thick. Thickness was uniform across the face of any single membrane used. The reactor was baffled, and was mixed by a six-bladed 1.75 in. turbine at 800 rpm, which ensured that no significant boundary layer resistance occurred at the exposed membrane surface. The fluid level was maintained constant by placing a suction tube at the desired level. Reactor ammonia concentration (and from that, the urea concentration remaining) was measured by drawing 50 ml samples of effluent every 10 min. The effluent sample was analyzed with an ammonia electrode-pH meter system. If necessary, reactor pH was maintained constant by dropwise addition of 1M H,SO,. Some 45-60 min were required for the reactor to achieve a steady state. After this occurred, and the effluent was sampled, the feed concentration of urea was changed and the procedure repeated. Reactor concentrations were generally in the range of 0.01 to 0.20M urea. After completion of a set of experiments the swollen membrane thickness was measured by a volume displacement technique, as was the swollen membrane mass. The membrane was then dried at 30°C overnight and reweighed. The mass of enzyme available for reaction was then determined from a simple material balance.
Membrane
Fig. 3. Continuous stirred-tank reactor with crosslinked urease-gelatin membrane in place.
2308
BOLLMEIER AND MIDDLEMAN
Figure 4 shows a typical set of data on reaction rate as a function of bulk urea concentration. The reaction rate, of course, follows simply from a steady-state material balance on the reactor, of the form
( Q / W ) ( s o- S B ) = r
(2)
where W is the mass of enzyme in the gelatin. One may immediately determine if the observed activity differs from that of urease in free solution, and Figure 4 shows the expected activity based on values of V , and K , taken from Figures 1 and 2. It is apparent that the activity of the immobilized enzyme has been reduced by an order of magnitude. DIFFUSION OF UREA THROUGH GELATIN
In order to evaluate the degree to which mass transfer resistance may account for the activity loss observed in Figure 4, it was first necessary to measure the diffusion coefficient of urea through a gelatin-urease crosslinked membrane. In order to measure diffusion alone it was necessary to eliminate the chemical reaction. This was done in two ways. The simplest method, of course, was to eliminate urease from the gelatin membrane altogether. However, the membrane-enzyme system was about 8% enzyme (on a dry weight basis), and so there was some concern that the enzyme might contribute a physical barrier to diffusion. Thus, a method of inactivating urease before its immobilization in gelatin was developed. With these two types of preparations, the diffusion coefficient of urea was measured as a function of pH and crosslinking time. Membranes were prepared as described earlier. In the diffusion studies, however, either the enzyme was omitted, or else the enzyme was inactivated. Inactivation began by making a solution of 54 mg p-chloromercuribenzoic acid in 6 to 10 drops of 10M NaOH, which was then mixed with 10 ml 0.1M TRIS at the desired pH. Then 10 mg n-ethylmaleimide were dissolved in 4 to 6 drops of ethyl alcohol, and this solution was combined with the first solution. To this final solution, 70 mg urease were added and the resulting solution was covered and periodically stirred at 41°C for 12 to 14 hr. The enzyme solution, now deactivated, was dialyzed for 2 hr against 500 ml O.1M TRIS. A diffusion cell was designed and constructed as shown in Figure 5. The membrane separates two well-mixed compartments. The top
HYDROLYSIS OF UREA
2309
I
I 0.I
S, ( m o le/ Ii t e r )
Fig. 4. Observed reaction rate vs. bulk stream urea concentration, at pH = 7. Curve is model based on free solution kinetic parameters at pH = 7. Membrane crosslinked 15 sec.
portion contains a urea solution of known initial concentration (usually 0.5 g/ml buffer) and fixed volume (usually 200 ml). The bottom section is a flow-through compartment for buffer alone, and the flow rate is high enough that the urea concentration in the bottom compartment is essentially zero. Urea concentration in the top cell was measured as a function of time. A simple quasi-steady analysis suggests that the diffusivity may be calculated from the relationship
(31 ln[c(t>/co] = - ( 9 A / l V ) t In eq. (3), A is the exposed surface of membrane that separates the two compartments and 1 is its thickness. V is the volume of the upper cell. Figure 6 shows a typical set of data from which 9may be calculated. The results of the diffusion experiments indicate no significant effect of pH, in the range 6 to 9, no significant effect of crosslinking
23 10
BOLLMEIER AND MIDDLEMAN
Fig. 5 .
Diffusion cell.
time in the range 15 to 30 sec, and no significant difference due to the presence or absence of inactive enzyme in the membrane. The measured value is taken to be 9 = (8.59 f 1.77) x cm2/sec. MODELING THE MASS TRANSFER RESISTANCE
Mass transfer may be accounted for by defining an effectiveness factor 7 as robs = qrsoln
t
(4)
(hr)
Fig. 6. Typical data using diffusion cell. The line gives 9 = 9.06 x A = 15.6 cm2; V = 200 cm3, 1 = 0.026 cm.
cm2/sec.
HYDROLYSIS OF UREA
2311
Here rsolnis the rate given by eq. (I), using the parameters V , and K , determined in free solution, and using the bulk reactor concentration for S. In order to determine the effectiveness factor it is necessary to solve the diffusion-reaction equation6,' within the membrane. Recalling that the membrane is backed on one side by an impermeable polyethylene sheet, this equation takes the dimensionless form (see Nomenclature) d28 8 -+2u--= 0 dz u+e with boundary conditions 8=1
atz=O
-d0 _ -0
atz=l dz Generally, eq. ( 5 ) must be solved numerically. However, for large Thiele modulus where the reaction is very rapid relative to diffusion, urea will not penetrate all the way to z = 1, and an alternate boundary condition,
+,
8=0
atz=l
allows an analytical solution for 8. The effectiveness factor then follows from?
and, using the approximate boundary condition noted above, 77 is found to be 17 =
(v'3/+)(1+ v)[v-'
-
In(1
+ v-')]"'
(7)
It follows, finally, that if the parameters V , and K , are unaffected by immobilization, the observed activity should be found to be robs = ( v'3v,/+)[v-'- In(l + v-1)]1'2 (8) Figure 7 compares this model with the data. It is clear that taking into account mass transfer resistance gives an improved model of the observed activity. This improvement is seen in other sets of data, as well, as Figures 8 and 9 show. Still, it is apparent that the mass transfer model fails at high conversion rates. A possible explanation lies in consideration of the pH within the membrane
2312
BOLLMEIER AND MIDDLEMAN
0
// 0
0
0.04
0.08
0.12
0.16
0.20
SB (mole/liter) Fig. 7. Same as Figure 4, but curve is model that accounts for internal diffusion, using free solution kinetic parameters at pH = 7.
pH PROFILE WITHIN THE MEMBRANE
When an enzymatic reaction gives products that can form ionic equilibria, it follows that the pH of the system could depend upon conversion. We may regard the enzymatic hydrolysis of urea to 4
X
L
0
S8 (mole/liter) Fig. 8. Observed reaction rate vs. bulk stream urea concentration, at pH = 8. Curves are models based o n free solution kinetic parameters at pH = 8. Upper curve is without internal diffusion, and lower curve is with diffusion accounted for. Membrane crosslinked 15 sec.
HYDROLYSIS O F UREA
2313
0.1
0
S, Fig. 9.
0.2
(mole/liter)
Same as Figure 8, but at pH
-
=
9.
obey the stoichiometric equation
(NH,),CO
+ H,O
urease
2 NH,
+ CO,
(9)
Ammonia and carbon dioxide form ionic equilibria, and the following reactions must be considered: CO, equilibria:
CO,
+ H 2 0 -+ H,CO, H,C03
$ H+
+ HC0,-
HCO,- ~2 H+ + C032-
There will be no dissolved CO, present at any pH used in our experiments. N H , equilibria:
Water equilibrium:
H 2 0 @ H+ + OH-
[H+][OH-]
=
K,
=
(13)
BOLLMEIER AND MIDDLEMAN
23 I4
TRIS equilibria. R NH,
+ H+ $ R NH3+
HzS04+ 2H+ +
= constant
[SO4’-] is known and constant, unless the buffering action of TRIS fails, and it is necessary to add significant amounts of H’SO,. These equations introduce nine unknown compositions: [R NHs’I, [C03z-l,
[R NHzI, “H4+1,
PI+], “H31,
[H CO3-I,
and
EHzCO31,
[OH-]
Written above are five equilibrium equations. Four more equations are needed. Three of these are material balances on TRIS buffer, CO,, and ammonia:
[CO,]o
+ [R NH3+] = [H CO3-] + [C03’-] + [HZCOSI
“H3Io
=
[R NHZ]o = [R NHz]
+
“H3I
S,
(15)
(16)
(17)
(rnole/liter)
Fig. 10. Predicted maximum and minimum pH within urease-gelatin membrane as a function of bulk stream urea concentration, at bulk pHB = 7 .
HYDROLYSIS OF UREA
2315
In eq. (15), [R NH,], refers to the known amount of TRIS buffer in the feed to the reactor. In eq. (16), [CO,], refers to the CO, produced by hydrolysis of urea. Consistent with the assumption that no dissolved CO, can exist at the operating pH levels studied, all COP dissociates. In eq. (17), [NH,], refers to the NH, produced by hydrolysis of urea. Since no CO, or NH, is fed to the reactor, it follows that [CO,], is related to [NH,], through the stoichiometry . [COZlO = f "H310. The ninth equation is supplied by a charge balance: 2[S04'-]
+ 2[C03'-] + [H CO3-I + [OH-] =
[H+] + [NH4+]+ [R NH,+]
(18)
Algebraic manipulation gives an implicit solution for [H+]:
(19) Equation (19) is taken to hold within the membrane, and it follows that the pH profile across the membrane is known if the profile of products, [CO,], and [NH,],, is known. But the product profiles must follow from simultaneous solution of
d2P + 2r(S, pH) = 0 dx
(21)
subject to surface conditions S=SB,
P = P B at
x=O
Here we use P to represent product NH,, and 9sand GBP are diffusivities of substrate (urea) and ammonia, respectively. In the surface conditions, S B and P B are the concentrations in the bulk solution external to the membrane. Both S B and P B are measured.
BOLLMEIER AND MIDDLEMAN
2316
Equations (20) and (21) may be combined to give
P(x)= PB
9 s + 2S B
9 a P
-
9 s
2 - S(x) % I
We may no longer regard S(x)as known from a solution of the simple diffusion-reaction equation [eq. ( 5 ) ] , because we no longer contend that the parameters K , and V , are independent of position through the membrane, since they depend on pH. Rather than deal with a very complex model at this point, it is possible to get some bounds on the maximum and minimum pH across the membrane. The minimum pH exists at the surface x = 0, where eq. (22) gives
P(0)= Pmin= PB
(23)
The maximum value is expected to occur at the surface x = 1. Assuming, as before, that the reaction is rapid relative to diffusion, so that S(1) = 0, we find (24) P ( 0 = pm,, = PB + 2@SDP)SB Equations (23) and (24) provide bounds on product NH, [the [NH,], term of eq. (19)]. From the stoichiometry, [COzl0= f[NH,I,, so bounds on this term are available as well. Thus we find bounds on [H+], the minimum being a t x = 0 and the maximum at x = I , by solving eq. (19). In eq. (24) the diffusivity ratio is taken as the same as that in water, which is known to be 9JgP= 0.73. Figures 10-12 show the results of calculations of maximum and
7
0.1
0
0.2
S B (rnole/liter) Fig. I I .
Same as Figure 10, but at pHB = 8.
2317
HYDROLYSIS OF UREA
I
9
min
PH
-
R
0
I 0.I
(
SB (mole/liter) Fig. 12. Same as Figure 10, but at pH,
=
9.
minimum pH. The minimum pH is not given by the bulk stream pH for all values of S , , and this is an artifact of the model that leads to eq. (19). It is assumed that [SO,2-] is constant, but TRIS becomes an ineffective buffer below pH = 7.3, and it was necessary to add more H,SO, as conversion increased, in order to keep the bulk pH constant. The maximum pH is the more significant parameter. At a bulk pH of 7, the enzyme would be nearly at its maximum activity, and as a consequence relatively large amounts of product would be formed and held within the gelatin membrane. As a consequence the maximum pH is predicted to be considerably in excess of that in the bulk stream. As Figure 1 shows, the activity drops precipitously as pH increases past pH = 7, and as a result one might expect to observe significant reduction in activity in the immobilized-enzyme experiment, due to this pH effect. Figures 11 and 12 shows similar, but smaller, effects. At pH = 9 more carbonate ion is produced than at lower pH. This releases another hydrogen ion and the system tends toward self-buffering. This is reflected in the curves of Figure 12. The next simplest stage Df modeling, then, is to take V, and K , from Figures 1 and 2, but use pH,,,, for each urea level, from Figures 10-12. The mass transfer resistance correction [eq. (S)] is used, with V, and Y (= K J S , ) calculated at pH,,,. These results are shown in Figures 13-15. In Figure 13 we see that the model gives a good representation of the data. At the higher pH values (Figs. 14 and 15) the representation is not quite so good. It should be kept in mind that the curves drawn in Figures 13-15 are based on pH,,,. The actual pH profile varies between pH, and pH,,,,
BOLLMEIER AND MIDDLEMAN
23 I8
-
0
0.04
0.12
0.08
0.16
0.20
S8 ( m o l e / l i t e r ) Fig. 13. Same as Figure 7, but curve is model that accounts for internal diffusion, using free solution kinetic parameters at pH,,,. Figure 10 is used to get pH,,, for each S.
and since pH > 7 is beyond the maximum for V,,,, a more exact theory would produce curves above the ones shown. Thus, in all cases, the data indicate activity that is less than that which can be accounted for by the mass transfer and pH effects discussed above. Thus we can conclude that there is, indeed, an intrinsic loss of activity associated with the immobilization procedure.
SB (mole/liter)
Fig. 14. Same as Figure 13, but pH,
=
8, and Figure 1 1 is used for pH,,,.
2319
HYDROLYSIS OF UREA
sB ( m o l e / l i t e r ) Fig. 15. Same as Figure 13, but pHs
=
9, and Figure 12 is used for
Figure 16 compares, at pH = 7, activity in two membranes that differ only in crosslinking time. There is a significant loss of activity at the higher crosslinking time, suggesting that glutaraldehyde interacts with the enzyme itself in a disadvantageous manner. Similar effects of longer crosslinking time are observed at pH 8 and 9, as well. CONCLUSIONS
The model outlined herein is incomplete, and so is approximate in several respects. For example, the pH-dependence of V , and
-
.F 3 -
-
0
I 0.I SB (mole/liter)
0.2
Fig. 16. Effect of crosslinking time on the observed reaction rate, all other 30 sec. variables held constant. Curve is same as in Fig. 13. ( 0 ) 15 sec, (0)
BOLLMEIER A N D MIDDLEMAN
2320
K , is taken to be that of the free stream data. Microenvironmental effects associated with crosslinking could alter this feature. In addition, of course, several approximations were made so as to allow analytical solutions of the diffusion-reaction equation. Thus it is not possible to determine the exact and individual degree of alteration of the urease activity associated with diffusion, immobilization, and pH distribution. We do conclude, however, that there are effects associated with each of these events, and we are able to estimate the degree of importance of each from the experiments described, and the corresponding approximate models. Nomenclature A C CO
9 K,
I P
Q r
S r Vrn W X
z
membrane cross-sectional area (cm2) concentration of urea in diffusion cell (mol/liter) initial value of c (moVliter) diffusivity of urea through gelatin (cm2/sec) Michaelis-Menten constant for free solution (mol/liter) membrane thickness (cm) product concentration (moVliter) reactor flow rate (cmVmin) reaction rate (mol substrateig enzyme.min) substrate (urea) concentration (moVliter) time maximum reaction rate for free solution (mol/g.min) enzyme mass ( g ) distance into membrane (cm) = XI1
Greek
PS
effectiveness factor = KJSB = I( V , / K r n 9 ) ” 2 ;Thiele modulus density of enzyme in membrane (g enzyme/cm3)
%
=
7) U
9 Subscript B
S/SB
refers t o values measured in bulk fluid external to membrane
References I . R. G. Carbonell and M. D. Kostin, AIChE J . , 18, I (1972). 2. W . R. Vieth and K. Venkatasubramanian, Chemrech., 677 (1973); ibid., 47, 309, 434 (1974). 3. J . M. Levine, R. Leon, and F. Stiegmann, Clin. C h e m . , 7 , 488 (1961). 4. K. B. Ramachandran and D. D. Perlmutter, Biorrchnol. Bioeng., 18, 685 (1976). 5 . P. V. Sundaram, Biochim. Biophys. Acra, 321, 319 (1973). 6. B. K. Hamilton, C. R. Gardner, and C . K. Colton, AIChE J . , 20, 503 (1974). 7. K. B. Bischoff, AIChE J . . 11, 351 (1965).
HYDROLYSIS OF UREA
232 I
8. J . M. Smith, Chemical Engineering Kinetics, 2nd ed. (McGraw-Hill, New York, 1970), pp. 427-439. 9. B. Atkinson, J. Rott, and I. Rousseau, Biotechnol. Bioeng., 19, 1037 (1977). 10. R. Goldman, 0. Kedem, and E. Katchalski, Biochemistry, 7, 4518 (1968). 11. L. Goldstein, Y. Levin, and E. Katchalski, Biochemistry, 3, 1913 (1964).
Accepted for Publication April 6, 1979
Summary Experiments and appropriate mathematical models are presented in an attempt to elucidate and separate the effects of mass transfer and immobilization on the apparent kinetics of hydrolysis of urea by urease immobilized within a crosslinked gelatin film. Diffusion of urea through the gelatin matrix appears to exert the major influence on the observed kinetics. Diffusion coefficients are measured, and a model for the “effectiveness factor” is presented, accounting for this aspect of mass transfer control. A secondary, but significant, influence on apparent kinetics arises because the reaction products lead to an increased pH level which, because of diffusion resistance, remains high within the gelatin matrix. For pH levels in the 6.7 to 9.0 range the activity of urease is a strongly decreasing function of pH. An approximate model accounting for ionic equilibrium allows this pH-diffusion effect to be introduced in such a way as to lead to predictions of the apparent kinetics that are compared with experimental observations. Examination of these results indicates that the immobilization procedure leads to some loss of activity due to an interaction of the gelatin crosslinking reaction with the enzyme itself.
INTRODUCTION
The potential of immobilized enzymes as catalysts in chemical reactors has motivated a considerable amount of research activity, and has led to the appearance of a significant amount of literature in this While the variety of enzymes, of carriers for immobilization, and suggested commercial applications confound the field somewhat, certain general features appear that serve as a link from one study to another: 1) Immobilization generally alters the activity of an enzyme relative to its free solution activity. 2) While apparent activity loss may occur due to mass transfer Biotechnology and Bioengineering, Vol. XXI, Pp. 2303-2321 (1979) 0006-3592/79/0021-2303$0 I .OO @ 1979 John Wiley & Sons, Inc.
2304
BOLLMEIER AND MIDDLEMAN
resistance to (or within) the solid carrier to (or within) which the enzyme is immobilized, infrinsic activity changes may occur as well. Changes in intrinsic activity (which we define to be those not explicable as simple effects of mass transfer resistance) may themselves arise from a variety of causes. For example, if immobilization is by covalent bonding to a carrier, the immobilization reaction may inactivate a certain fraction of the enzyme molecules. It is also possible that upon immobilization the immediate surroundings (the microenvironment) of the enzyme may alter their configuration sufficiently to alter activity. This latter effect may occur whether the immobilization is through covalent bonding, or more simply through physical entrapment within the carrier matrix. There is yet another means of activity alteration, but one which in a sense is a combination of mass transfer and microenvironmental effects. If the reaction product has the capacity to alter the reaction rate (two examples would be product inhibition and pH dependence of activity), then immobilization can produce an environment surrounding the enzyme which, because of mass transfer resistance, has a composition different from the surrounding solution. Thus, the enzyme would exhibit a different activity than in free solution. If one wishes to design an immobilized-enzyme reactor, it is necessary to be able to separate these effects on activity, and to determine the conditions under which each of these effects will occur and the extent to which the apparent activity is altered. This paper reports a study of urease immobilized within crosslinked gelatin. Through a combination of experiments and modeling we have been able to determine the separate contributions of the phenomena described above. KINETICS IN FREE SOLUTION
Enzymatic reactions are often observed to obey the MichaelisMenten model, in which the reaction rate is given by
r
=
V,S/(K,
+ S)
Here, S refers to the concentration of reactant (called “substrate” in the biochemical literature). For small concentrations of substrate the kinetics appear to be first order. For large S the kinetics approach zeroth order, and V , is the “maximum velocity” of the reaction. V,, and thus r , are based on unit mass of enzyme, so r is the conversion rate per unit mass of enzyme.
HYDROLYSIS OF UREA
2305
The Michaelis-Menten parameters were measured in our studies from initial rates of reaction carried out at constant temperature in a well-stirred batch reactor. At a given pH value (5 5 pH 5 9) a set of urea solutions was prepared by adding from 0.1 to 0.9 g urea to 100 ml buffer [O.lM TRIS (hydroxymethyl) amino methane]. Each urea solution was then diluted in 200 ml O.1M TRIS with 1mM EDTA, adjusted to the desired pH with sulfuric acid. This solution was in a stirred beaker in a thermostated bath. To this solution, 2 ml 0.1 M TRIS with 1mM EDTA containing 11.7 mg urease (Sigma type 111, lot No. 66c-7550) were added. One ml samples were withdrawn at 1 min intervals for 11 min, and the samples were analyzed for urea by the Levine3 method, using a Leitz model M photometer. Temperature was controlled at 25°C. pH was measured continuously and maintained constant by dropwise addition of 1M H,SO, when necessary. For each experiment the initial rate was determined from the slope of urea concentration versus time data. The K , and V , parameters were then determined by fitting the Michaelis- Menten model to the data. Figures 1 and 2 show the results. A maximum in V , at pH = 6.75 is seen, which is consistent with published results for ~ r e a s e . The ~ , ~ K , values were subject to considerable uncertainty for pH 6 to 7, and it is not clear if a true maximum exists, as the data suggest.
0 4
5
6
?
a
9
PH
Fig. 1. 25°C.
Maximum rate velocity as a function of pH for urease in free solution, at
2306
BOLLMEIER A N D MIDDLEMAN
? 01 5
0 0
I 6
1
I
7
8
a +
9
PH
Fig. 2. Michaelis-Menten constant, K , , as a function of pH for urease in free solution, at 25°C.
OBSERVED IMMOBILIZED UREASE KINETICS
Urease-gelatin membranes were formed by first dissolving 0.8 g pigskin gelatin (Swift’s lot No. 3884-2) in 10 ml O.1M TRIS with ImM EDTA at 35°C. To this solution 70 mg urease were added, and after complete dissolution the solution was poured onto a stretched polyethylene film to form a puddle about 8 cm in diam. Once gelled, the membrane was placed in a forced-air oven and allowed to dry overnight at 27°C. The dried membrane thickness was measured with a micrometer, and then the membrane was swollen by placing it in O.1M TRIS with ImM EDTA, at the desired pH, for 15 min at 5°C. The swollen membrane was then crosslinked in a 10% solution of glutaraldehyde for 15 or 30 sec, depending on the desired degree of crosslinking. The freshly crosslinked membrane was washed well with deionized distilled water before use in the reactor. Figure 3 shows the reactor that was fabricated of acrylic pipe and sheet and had a volume of approximately 70 ml. For the kinetic studies the reactor was operated as a continuous-flow stirred tank at 25°C. The membrane was cut to a 67 mm diam and backed by an identical piece of polyethylene for support. The area of membrane exposed to reactant was 15.55 cm2. Reactor feed was from a stain-
HYDROLYSIS OF UREA
2307
less-steel pressure vessel. The feed stock was a urea solution in 0.1M TRIS with 1mM EDTA. The feed rate was 14 ml/min. Individual membranes varied in wet thickness, but were generaily from 0.5 to 1 mm thick. Thickness was uniform across the face of any single membrane used. The reactor was baffled, and was mixed by a six-bladed 1.75 in. turbine at 800 rpm, which ensured that no significant boundary layer resistance occurred at the exposed membrane surface. The fluid level was maintained constant by placing a suction tube at the desired level. Reactor ammonia concentration (and from that, the urea concentration remaining) was measured by drawing 50 ml samples of effluent every 10 min. The effluent sample was analyzed with an ammonia electrode-pH meter system. If necessary, reactor pH was maintained constant by dropwise addition of 1M H,SO,. Some 45-60 min were required for the reactor to achieve a steady state. After this occurred, and the effluent was sampled, the feed concentration of urea was changed and the procedure repeated. Reactor concentrations were generally in the range of 0.01 to 0.20M urea. After completion of a set of experiments the swollen membrane thickness was measured by a volume displacement technique, as was the swollen membrane mass. The membrane was then dried at 30°C overnight and reweighed. The mass of enzyme available for reaction was then determined from a simple material balance.
Membrane
Fig. 3. Continuous stirred-tank reactor with crosslinked urease-gelatin membrane in place.
2308
BOLLMEIER AND MIDDLEMAN
Figure 4 shows a typical set of data on reaction rate as a function of bulk urea concentration. The reaction rate, of course, follows simply from a steady-state material balance on the reactor, of the form
( Q / W ) ( s o- S B ) = r
(2)
where W is the mass of enzyme in the gelatin. One may immediately determine if the observed activity differs from that of urease in free solution, and Figure 4 shows the expected activity based on values of V , and K , taken from Figures 1 and 2. It is apparent that the activity of the immobilized enzyme has been reduced by an order of magnitude. DIFFUSION OF UREA THROUGH GELATIN
In order to evaluate the degree to which mass transfer resistance may account for the activity loss observed in Figure 4, it was first necessary to measure the diffusion coefficient of urea through a gelatin-urease crosslinked membrane. In order to measure diffusion alone it was necessary to eliminate the chemical reaction. This was done in two ways. The simplest method, of course, was to eliminate urease from the gelatin membrane altogether. However, the membrane-enzyme system was about 8% enzyme (on a dry weight basis), and so there was some concern that the enzyme might contribute a physical barrier to diffusion. Thus, a method of inactivating urease before its immobilization in gelatin was developed. With these two types of preparations, the diffusion coefficient of urea was measured as a function of pH and crosslinking time. Membranes were prepared as described earlier. In the diffusion studies, however, either the enzyme was omitted, or else the enzyme was inactivated. Inactivation began by making a solution of 54 mg p-chloromercuribenzoic acid in 6 to 10 drops of 10M NaOH, which was then mixed with 10 ml 0.1M TRIS at the desired pH. Then 10 mg n-ethylmaleimide were dissolved in 4 to 6 drops of ethyl alcohol, and this solution was combined with the first solution. To this final solution, 70 mg urease were added and the resulting solution was covered and periodically stirred at 41°C for 12 to 14 hr. The enzyme solution, now deactivated, was dialyzed for 2 hr against 500 ml O.1M TRIS. A diffusion cell was designed and constructed as shown in Figure 5. The membrane separates two well-mixed compartments. The top
HYDROLYSIS OF UREA
2309
I
I 0.I
S, ( m o le/ Ii t e r )
Fig. 4. Observed reaction rate vs. bulk stream urea concentration, at pH = 7. Curve is model based on free solution kinetic parameters at pH = 7. Membrane crosslinked 15 sec.
portion contains a urea solution of known initial concentration (usually 0.5 g/ml buffer) and fixed volume (usually 200 ml). The bottom section is a flow-through compartment for buffer alone, and the flow rate is high enough that the urea concentration in the bottom compartment is essentially zero. Urea concentration in the top cell was measured as a function of time. A simple quasi-steady analysis suggests that the diffusivity may be calculated from the relationship
(31 ln[c(t>/co] = - ( 9 A / l V ) t In eq. (3), A is the exposed surface of membrane that separates the two compartments and 1 is its thickness. V is the volume of the upper cell. Figure 6 shows a typical set of data from which 9may be calculated. The results of the diffusion experiments indicate no significant effect of pH, in the range 6 to 9, no significant effect of crosslinking
23 10
BOLLMEIER AND MIDDLEMAN
Fig. 5 .
Diffusion cell.
time in the range 15 to 30 sec, and no significant difference due to the presence or absence of inactive enzyme in the membrane. The measured value is taken to be 9 = (8.59 f 1.77) x cm2/sec. MODELING THE MASS TRANSFER RESISTANCE
Mass transfer may be accounted for by defining an effectiveness factor 7 as robs = qrsoln
t
(4)
(hr)
Fig. 6. Typical data using diffusion cell. The line gives 9 = 9.06 x A = 15.6 cm2; V = 200 cm3, 1 = 0.026 cm.
cm2/sec.
HYDROLYSIS OF UREA
2311
Here rsolnis the rate given by eq. (I), using the parameters V , and K , determined in free solution, and using the bulk reactor concentration for S. In order to determine the effectiveness factor it is necessary to solve the diffusion-reaction equation6,' within the membrane. Recalling that the membrane is backed on one side by an impermeable polyethylene sheet, this equation takes the dimensionless form (see Nomenclature) d28 8 -+2u--= 0 dz u+e with boundary conditions 8=1
atz=O
-d0 _ -0
atz=l dz Generally, eq. ( 5 ) must be solved numerically. However, for large Thiele modulus where the reaction is very rapid relative to diffusion, urea will not penetrate all the way to z = 1, and an alternate boundary condition,
+,
8=0
atz=l
allows an analytical solution for 8. The effectiveness factor then follows from?
and, using the approximate boundary condition noted above, 77 is found to be 17 =
(v'3/+)(1+ v)[v-'
-
In(1
+ v-')]"'
(7)
It follows, finally, that if the parameters V , and K , are unaffected by immobilization, the observed activity should be found to be robs = ( v'3v,/+)[v-'- In(l + v-1)]1'2 (8) Figure 7 compares this model with the data. It is clear that taking into account mass transfer resistance gives an improved model of the observed activity. This improvement is seen in other sets of data, as well, as Figures 8 and 9 show. Still, it is apparent that the mass transfer model fails at high conversion rates. A possible explanation lies in consideration of the pH within the membrane
2312
BOLLMEIER AND MIDDLEMAN
0
// 0
0
0.04
0.08
0.12
0.16
0.20
SB (mole/liter) Fig. 7. Same as Figure 4, but curve is model that accounts for internal diffusion, using free solution kinetic parameters at pH = 7.
pH PROFILE WITHIN THE MEMBRANE
When an enzymatic reaction gives products that can form ionic equilibria, it follows that the pH of the system could depend upon conversion. We may regard the enzymatic hydrolysis of urea to 4
X
L
0
S8 (mole/liter) Fig. 8. Observed reaction rate vs. bulk stream urea concentration, at pH = 8. Curves are models based o n free solution kinetic parameters at pH = 8. Upper curve is without internal diffusion, and lower curve is with diffusion accounted for. Membrane crosslinked 15 sec.
HYDROLYSIS O F UREA
2313
0.1
0
S, Fig. 9.
0.2
(mole/liter)
Same as Figure 8, but at pH
-
=
9.
obey the stoichiometric equation
(NH,),CO
+ H,O
urease
2 NH,
+ CO,
(9)
Ammonia and carbon dioxide form ionic equilibria, and the following reactions must be considered: CO, equilibria:
CO,
+ H 2 0 -+ H,CO, H,C03
$ H+
+ HC0,-
HCO,- ~2 H+ + C032-
There will be no dissolved CO, present at any pH used in our experiments. N H , equilibria:
Water equilibrium:
H 2 0 @ H+ + OH-
[H+][OH-]
=
K,
=
(13)
BOLLMEIER AND MIDDLEMAN
23 I4
TRIS equilibria. R NH,
+ H+ $ R NH3+
HzS04+ 2H+ +
= constant
[SO4’-] is known and constant, unless the buffering action of TRIS fails, and it is necessary to add significant amounts of H’SO,. These equations introduce nine unknown compositions: [R NHs’I, [C03z-l,
[R NHzI, “H4+1,
PI+], “H31,
[H CO3-I,
and
EHzCO31,
[OH-]
Written above are five equilibrium equations. Four more equations are needed. Three of these are material balances on TRIS buffer, CO,, and ammonia:
[CO,]o
+ [R NH3+] = [H CO3-] + [C03’-] + [HZCOSI
“H3Io
=
[R NHZ]o = [R NHz]
+
“H3I
S,
(15)
(16)
(17)
(rnole/liter)
Fig. 10. Predicted maximum and minimum pH within urease-gelatin membrane as a function of bulk stream urea concentration, at bulk pHB = 7 .
HYDROLYSIS OF UREA
2315
In eq. (15), [R NH,], refers to the known amount of TRIS buffer in the feed to the reactor. In eq. (16), [CO,], refers to the CO, produced by hydrolysis of urea. Consistent with the assumption that no dissolved CO, can exist at the operating pH levels studied, all COP dissociates. In eq. (17), [NH,], refers to the NH, produced by hydrolysis of urea. Since no CO, or NH, is fed to the reactor, it follows that [CO,], is related to [NH,], through the stoichiometry . [COZlO = f "H310. The ninth equation is supplied by a charge balance: 2[S04'-]
+ 2[C03'-] + [H CO3-I + [OH-] =
[H+] + [NH4+]+ [R NH,+]
(18)
Algebraic manipulation gives an implicit solution for [H+]:
(19) Equation (19) is taken to hold within the membrane, and it follows that the pH profile across the membrane is known if the profile of products, [CO,], and [NH,],, is known. But the product profiles must follow from simultaneous solution of
d2P + 2r(S, pH) = 0 dx
(21)
subject to surface conditions S=SB,
P = P B at
x=O
Here we use P to represent product NH,, and 9sand GBP are diffusivities of substrate (urea) and ammonia, respectively. In the surface conditions, S B and P B are the concentrations in the bulk solution external to the membrane. Both S B and P B are measured.
BOLLMEIER AND MIDDLEMAN
2316
Equations (20) and (21) may be combined to give
P(x)= PB
9 s + 2S B
9 a P
-
9 s
2 - S(x) % I
We may no longer regard S(x)as known from a solution of the simple diffusion-reaction equation [eq. ( 5 ) ] , because we no longer contend that the parameters K , and V , are independent of position through the membrane, since they depend on pH. Rather than deal with a very complex model at this point, it is possible to get some bounds on the maximum and minimum pH across the membrane. The minimum pH exists at the surface x = 0, where eq. (22) gives
P(0)= Pmin= PB
(23)
The maximum value is expected to occur at the surface x = 1. Assuming, as before, that the reaction is rapid relative to diffusion, so that S(1) = 0, we find (24) P ( 0 = pm,, = PB + 2@SDP)SB Equations (23) and (24) provide bounds on product NH, [the [NH,], term of eq. (19)]. From the stoichiometry, [COzl0= f[NH,I,, so bounds on this term are available as well. Thus we find bounds on [H+], the minimum being a t x = 0 and the maximum at x = I , by solving eq. (19). In eq. (24) the diffusivity ratio is taken as the same as that in water, which is known to be 9JgP= 0.73. Figures 10-12 show the results of calculations of maximum and
7
0.1
0
0.2
S B (rnole/liter) Fig. I I .
Same as Figure 10, but at pHB = 8.
2317
HYDROLYSIS OF UREA
I
9
min
PH
-
R
0
I 0.I
(
SB (mole/liter) Fig. 12. Same as Figure 10, but at pH,
=
9.
minimum pH. The minimum pH is not given by the bulk stream pH for all values of S , , and this is an artifact of the model that leads to eq. (19). It is assumed that [SO,2-] is constant, but TRIS becomes an ineffective buffer below pH = 7.3, and it was necessary to add more H,SO, as conversion increased, in order to keep the bulk pH constant. The maximum pH is the more significant parameter. At a bulk pH of 7, the enzyme would be nearly at its maximum activity, and as a consequence relatively large amounts of product would be formed and held within the gelatin membrane. As a consequence the maximum pH is predicted to be considerably in excess of that in the bulk stream. As Figure 1 shows, the activity drops precipitously as pH increases past pH = 7, and as a result one might expect to observe significant reduction in activity in the immobilized-enzyme experiment, due to this pH effect. Figures 11 and 12 shows similar, but smaller, effects. At pH = 9 more carbonate ion is produced than at lower pH. This releases another hydrogen ion and the system tends toward self-buffering. This is reflected in the curves of Figure 12. The next simplest stage Df modeling, then, is to take V, and K , from Figures 1 and 2, but use pH,,,, for each urea level, from Figures 10-12. The mass transfer resistance correction [eq. (S)] is used, with V, and Y (= K J S , ) calculated at pH,,,. These results are shown in Figures 13-15. In Figure 13 we see that the model gives a good representation of the data. At the higher pH values (Figs. 14 and 15) the representation is not quite so good. It should be kept in mind that the curves drawn in Figures 13-15 are based on pH,,,. The actual pH profile varies between pH, and pH,,,,
BOLLMEIER AND MIDDLEMAN
23 I8
-
0
0.04
0.12
0.08
0.16
0.20
S8 ( m o l e / l i t e r ) Fig. 13. Same as Figure 7, but curve is model that accounts for internal diffusion, using free solution kinetic parameters at pH,,,. Figure 10 is used to get pH,,, for each S.
and since pH > 7 is beyond the maximum for V,,,, a more exact theory would produce curves above the ones shown. Thus, in all cases, the data indicate activity that is less than that which can be accounted for by the mass transfer and pH effects discussed above. Thus we can conclude that there is, indeed, an intrinsic loss of activity associated with the immobilization procedure.
SB (mole/liter)
Fig. 14. Same as Figure 13, but pH,
=
8, and Figure 1 1 is used for pH,,,.
2319
HYDROLYSIS OF UREA
sB ( m o l e / l i t e r ) Fig. 15. Same as Figure 13, but pHs
=
9, and Figure 12 is used for
Figure 16 compares, at pH = 7, activity in two membranes that differ only in crosslinking time. There is a significant loss of activity at the higher crosslinking time, suggesting that glutaraldehyde interacts with the enzyme itself in a disadvantageous manner. Similar effects of longer crosslinking time are observed at pH 8 and 9, as well. CONCLUSIONS
The model outlined herein is incomplete, and so is approximate in several respects. For example, the pH-dependence of V , and
-
.F 3 -
-
0
I 0.I SB (mole/liter)
0.2
Fig. 16. Effect of crosslinking time on the observed reaction rate, all other 30 sec. variables held constant. Curve is same as in Fig. 13. ( 0 ) 15 sec, (0)
BOLLMEIER A N D MIDDLEMAN
2320
K , is taken to be that of the free stream data. Microenvironmental effects associated with crosslinking could alter this feature. In addition, of course, several approximations were made so as to allow analytical solutions of the diffusion-reaction equation. Thus it is not possible to determine the exact and individual degree of alteration of the urease activity associated with diffusion, immobilization, and pH distribution. We do conclude, however, that there are effects associated with each of these events, and we are able to estimate the degree of importance of each from the experiments described, and the corresponding approximate models. Nomenclature A C CO
9 K,
I P
Q r
S r Vrn W X
z
membrane cross-sectional area (cm2) concentration of urea in diffusion cell (mol/liter) initial value of c (moVliter) diffusivity of urea through gelatin (cm2/sec) Michaelis-Menten constant for free solution (mol/liter) membrane thickness (cm) product concentration (moVliter) reactor flow rate (cmVmin) reaction rate (mol substrateig enzyme.min) substrate (urea) concentration (moVliter) time maximum reaction rate for free solution (mol/g.min) enzyme mass ( g ) distance into membrane (cm) = XI1
Greek
PS
effectiveness factor = KJSB = I( V , / K r n 9 ) ” 2 ;Thiele modulus density of enzyme in membrane (g enzyme/cm3)
%
=
7) U
9 Subscript B
S/SB
refers t o values measured in bulk fluid external to membrane
References I . R. G. Carbonell and M. D. Kostin, AIChE J . , 18, I (1972). 2. W . R. Vieth and K. Venkatasubramanian, Chemrech., 677 (1973); ibid., 47, 309, 434 (1974). 3. J . M. Levine, R. Leon, and F. Stiegmann, Clin. C h e m . , 7 , 488 (1961). 4. K. B. Ramachandran and D. D. Perlmutter, Biorrchnol. Bioeng., 18, 685 (1976). 5 . P. V. Sundaram, Biochim. Biophys. Acra, 321, 319 (1973). 6. B. K. Hamilton, C. R. Gardner, and C . K. Colton, AIChE J . , 20, 503 (1974). 7. K. B. Bischoff, AIChE J . . 11, 351 (1965).
HYDROLYSIS OF UREA
232 I
8. J . M. Smith, Chemical Engineering Kinetics, 2nd ed. (McGraw-Hill, New York, 1970), pp. 427-439. 9. B. Atkinson, J. Rott, and I. Rousseau, Biotechnol. Bioeng., 19, 1037 (1977). 10. R. Goldman, 0. Kedem, and E. Katchalski, Biochemistry, 7, 4518 (1968). 11. L. Goldstein, Y. Levin, and E. Katchalski, Biochemistry, 3, 1913 (1964).
Accepted for Publication April 6, 1979
