Langmuir 2002, 18, 5661-5667
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Articles Enzymatic Formation of pH Gradients within Polyelectrolyte Gels with Immobilized Urease Kazuyoshi Ogawa and Etsuo Kokufuta* Institute of Applied Biochemistry, University of Tsukuba, Tsukuba, Ibaraki 305-8572, Japan Received October 5, 2001. In Final Form: May 6, 2002 Thermally responsive cationic gels with immobilized urease, in the shape of a small cylinder with a diameter of 290-640 µm, were prepared via gelation of an aqueous monomer solution containing the enzyme. N-Isopropylacrylamide and N-vinylimidazole were used as thermosensitive and pH-sensitive monomers, respectively. Diameters at different positions of the cylinder were microscopically measured in a cell through which substrate solution (pH 4; 35 °C) was passed at a constant flow rate; thus, both substrate concentration and pH at the gel surface were maintained at a constant level throughout the experimental period. It was found that the gel undergoes a shrinking change due to an enzymatically induced increase in pH within the gel phase. There was a marked position dependence of the shrinking degree; the diameter at the center of the cylinder was smaller than that at either the top or the bottom, but the diameters at the top and bottom were identical with each other. This trend was observed during the period of shrinking of gel and after the establishment of swelling equilibrium of gel. To account for these results in connection with a pH gradient, which would be enzymatically formed within the gel phase, mathematical simulations were conducted with a reaction-diffusion model. The central part of the gel was then taken as an infinitely long circular cylinder. There was a good agreement between the results of simulations and experiments after equilibrium swelling was reached. Therefore, it is reasonable to conclude that (i) a charge distribution depending on the pH gradient would appear in the gel and (ii) this distribution affects the overall swelling ratio of the gel. These can no longer be explained in terms of the concept of osmotic pressure arising from mobile counterions within the gel phase.
Introduction As early as 1948, Vermaas and Hermans1 studied the behavior of lightly cross-linked polymer chain networks with ionic charges, that is, polyelectrolyte gels. Shortly afterward Flory2 and Katchalsky3 published pioneering works on theories and experiments of polyelectrolyte gels. In these early studies, however, little attention was paid to the existence of a critical endpoint in the phase equilibria, although in 1968 Dusek and Patterson4 published a theoretical paper on this subject. The earliest report of phase transition in a neutral gel was in 1978 by Tanaka,5 who discovered a discontinuous volume collapse of poly(acrylamide) (PAAm) gels in acetone-water mixtures when varying the temperature or the composition of the mixture. On the phase transition in polyelectrolyte gels, two studies by Tanaka and his colleagues using partially hydrolyzed PAAm gels6 and copolymer gels of acrylic acid (AAc) with N-isopropylacrylamide (NIPA)7 would be of historical importance. To account for the phase transition theoretically, they modified Flory’s formula in * Author to whom correspondence should be addressed (fax 81-298-53-4605; e-mail
[email protected]). (1) Vermaas, D.; Hermans, J. J. Recueil 1948, 67, 983. (2) Flory P. J. Principle of Polymer Chemistry; Cornell University Press: New York, 1953; pp 576-593. (3) Katchalsky, A.; Michaeli, I. J. Polym. Sci. 1955, 15, 69. (4) Dusek, K.; Patterson, D. J. Polym. Sci., Polym. Phys. Ed. 1968, 6, 1209. (5) Tanaka, T. Phys. Rev. Lett. 1978, 12, 820. (6) Tanaka, T.; Fillmore, D. J.; Sun, S.-T.; Nishio, I.; Swislow, G.; Shah, A. Phys. Rev. Lett. 1980, 45, 1636. (7) Hirotsu, S.; Hirokawa, Y.; Tanaka, T. J. Chem. Phys. 1987, 87, 1392.
which the osmotic pressure of a gel was written as a sum of three contributions from polymer-solvent mixing (mixing term), rubber elasticity (elastic term), and a difference in the concentration of mobile ions between the gel and the bulk phase (ionic term). At present there are many arguments against the use of Flory’s formula, so alternative descriptions have been made for the mixing and elastic terms (e.g., see refs 8 and 9). For the ionic term, we must return to the work by Katchalsky et al.,3 who argued that Donnan equilibria assumed by Flory2 could not give even a rough approximation. Then they proposed a theory based on a random coil model in which the charge interaction along a chain was considered on the basis of Debye-Hu¨ckel theory. Their work was taken over by Hasa-Ilavsky-Dusek (HID) theory10 in which a modification of the elastic term was made by considering the influence of charges on the deformation of the network. Although there was such a historical controversy, the “concept” of osmotic pressure arising from mobile ions has been employed in the understanding of the nature of the swelling or the phase transition in polyelectrolyte gels. It is preferable to verify experimentally whether the above concept is adequate or not, especially in view of both the scientific interest and the technological significance of polyelectrolyte gels. This is, however, a rather difficult problem with respect to experimental techniques. (8) Schild, H. G. Prog. Polym. Sci. 1992, 17, 163. (9) Kokufuta, E. Phase Transtions in Polyelectrolyte Gels. In Physical Chemistry of Polyelectrolytes; Radeva, T., Ed.;Dekker: New York, 2000; pp 591-664. (10) Hasa, J.; Ilavsky, M.; Dusek, K. J. Polym. Sci., Polym. Phys. Ed. 1975, 13, 253.
10.1021/la0115177 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/25/2002
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Thus, to our knowledge, only one paper11 has attempted to directly compare Flory’s formula as modified by Tanaka and the formula based on HID theory, but the conclusion drawn was that neither the former nor the latter can provide a reasonable approach for predicting the swelling degree of polyelectrolyte gels. In contrast to such a direct comparison, the use of ionic gels with different distributions of network-bound charges would provide a key to discussion about the role of ionic charges in the swelling of polyelectrolyte gels. If the concept of osmotic pressure arising from mobile ions is correct, one might not observe any difference in the swelling curves between the two kinds of ionic gels into which the fixed charges were homogeneously and inhomogeneously introduced. The reason is that counterions to the ionized groups should move freely within the gel phase surrounded by the Donnan potential barrier and thereby increase the osmotic pressure acting to swell the gel. Taking the above into account, we have studied the swelling behavior of ionic gels with inhomogeneous charge distributions.12,13 A gel sample was prepared via binding of sodium dodecylbenzenesulfonate to lightly cross-linked NIPA polymer networks;12 another was based on the physical entrapment of poly(AAc) within the NIPA chain network.13 Then we observed a strong effect of the charge inhomogeneity on the swelling degree of both ionic gels, indicating that the swelling behavior can no longer be explained in terms of the concept of osmotic pressure arising from mobile counterions within the gel phase. To generalize this conclusion, it is necessary to develop other techniques for establishing charge inhomogeneities in the gel. For this purpose, we studied here an enzymatic approach for forming a pH gradient to establish an inhomogeneous distribution of charges within the gel phase. This paper reports the fact that a positiondependent change of the swelling degree appears in a small NIPA-based cylindrical gel containing imidazole groups and urease while the following enzyme reaction14 takes place:
A detailed analysis of experimental data was performed by comparing them with the results of mathematical simulations with a reaction-diffusion model. It was found that the observed position dependence is explicable in terms of the enzymatic formation of pH gradients within the gel, the aspect of which can be clarified in the mathematical simulation. Experimental Section Thermally responsive ionic gels with immobilized urease were prepared via gelation of aqueous pregel solutions with the following compositions (in milligrams per milliliter of pure (11) Tong, Z.; Liu, X. Macromolecules 1994, 27, 844. (12) Kokufuta, E.; Suzuki, H.; Sakamoto, D. Langmuir 1997, 13, 2627. (13) Kokufuta, E.; Wang, B.; Yoshida, R.; Khokhlov, A. R.; Hirata, M. Macromolecules 1998, 31, 6878. (14) Reithel, F. J. Ureases. In The Enzymes, 3rd ed.; Boyer, P. D., Ed.; Academic Press: New York, 1971; Vol. 4, pp 1-21.
Ogawa and Kokufuta water): NIPA (thermosensitive monomer), 71.3; N-vinylimidazole (pH-sensitive ionic monomer), 6.59; N,N′-methylenebis(acrylamide) (cross-linker), 1.33; and urease (jack bean source), 0.28.0 mg. The polymerization was initiated by a pair of ammonium persulfate (APS; initiator) and N,N,N′,N′-tetramethylethylenediamine (TMED; accelerator), the amounts of which were 0.4 mg (APS) and 4.8 µL (TMED) per milliliter of pregel solution. The gel samples, cylindrical in shape, were obtained in a test tube into which capillaries with different inner diameters (290-637 µm) had been inserted prior to gelation. All of the samples were taken out of the capillaries, purified by repeated swelling and shrinking procedures, cut into cylinders with a diameter/height ratio ) ca. 1:4, and then stored at 3 °C before use. The overall charge (C0, 0.775 ( 0.09 mmol/g of dry polymer) of our gels was determined by potentiometric titration at ionic strength 0.015 (NaCl) and at 35 °C. For the titration the gel samples (of a fine grind) with 0.4 and 8 mg/mL of the immobilized enzyme were taken out of the test tubes from which the capillaries had been drawn out, purified as described above, and converted into the salt form. Diameters at different positions on the height of a gel cylinder were microscopically measured at 35 °C and at suitable time intervals in a flow (1 mL/min) of the substrate solution, that is, 5 mM maleate buffer (pH 4) containing 1 mM urea. The details of apparatus and measuring procedures have been described in full in our previous papers.15,16 Kinetic studies of urease-catalyzed hydrolysis of urea were performed at 35 °C and at different pH values (5.5, 6.5, and 7.0) according to a method of Ambrose et al.17 To estimate both maximum velocity and Michaelis constant at each pH by means of Lineweaver-Burk plots, the activity was measured at various concentrations of the substrate solution, which was prepared with the same buffer system as used in the size measurement. In addition, we performed measurements of the pH and ammonia concentration at different stages of the enzyme reaction using the maleate buffer, the initial pH of which was adjusted to 4.0. The ammonia concentration was colorimetrically determined with Nessler’s reagent.
Results Figure 1 shows typical time courses of enzymatically and chemically induced diameter changes of a cylindrical gel with immobilized urease. The enzymatic and chemical means of shrinking the gel were based, respectively, on the use of a substrate-containing buffer (pH 4) and of a substrate-free buffer (pH 6 with NH4OH). We measured diameters at the top (d1), center (d2), and bottom (d3) of the cylindrical gel. It was observed that in the chemical gel collapse the diameters at these three positions are identical with one another, particularly in an equilibrium state at which the swelling ratio is independent of time. In the enzymatic process, there is no difference in the diameters at the top and at the bottom, but values at both positions are larger than that at the center after the establishment of swelling equilibrium. Thus, the enzymatic shrinking allows the gel to change its shape from that of a “real” cylinder to that of a “distorted” cylinder with the top and bottom still in a swollen state. However, this shape change was not observed when the amount of the immobilized enzyme became very high (because the enzyme reaction takes place only at the liquid-gel interface; see Discussion). The present cationic gels undergo a shrinking change when the pH is increased, because the network charges are eliminated via the dissociation of protons from the imidazole ions (dNH+-). The immobilized enzyme reac(15) Kokufuta, E.; Suzuki, H.; Yoshida, R.; Yamada, K.; Hirata, M.; Kaneko, F. Langmuir 1998, 14, 795. (16) Kokufuta, E.; Suzuki, H.; Yoshida, R.; Kaneko, F.; Yamada, K.; Hirata, M. Colloids Surf. 1999, A147, 179. (17) Ambrose, J. F.; Kistiakowsky, G. B.; Kridl, A. G. J. Am. Chem. Soc. 1950, 72, 317.
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Figure 2. Long cylinder model with radius R for analyzing the transient variations of substrate and product concentrations within the central part of our gel system. Substrate molecules diffuse into the cylindrical gel through its lateral surface at which substrate concentration ([S]) is kept constant at [S]0. The products (ammonium bicarbonate in this case) enzymatically formed within the gel phase diffuse into the outer medium, but at the lateral surface the product concentration ([P]0) is maintained at zero due to a continuous sweeping of the gel surface with a stream of the substrate solution.
Figure 1. Time-dependent changes in diameters at the top (d1), center (d2), and bottom (d3) of cylindrical gel with immobilized urease: (a) enzymatic shrinking, which was made through quick replacement of the outer medium (5 mM maleate buffer not containing the substrate; pH 4) by the substrate solution (5 mM maleate buffer containing 1 mM urea as the substrate; pH 4); (b) chemical shrinking made through quick replacement of the outer medium (5 mM maleate buffer, pH 4) by another medium (5 mM maleate buffer, pH 6). We used the same gel in measurements a and b, so that P-1 denotes the gel before the replacements of the outer medium by the substrate solution or by the substrate-free buffer. P-2 is the gel in the substrate solution after 60 min, and P-3 is the gel in the substrate-free medium at pH 6 after 60 min. Measurements were carried out at 35 °C in a flow (1 mL/min) of the outer solutions.
tion in eq 1 raises pH within the gel phase; therefore, the gel shrinks. Such an enzymatic increase in pH occurs by diffusion of the substrate from the outer solution, but at the same time the reaction product diffuses into the outer solution. These conditions permit a pH gradient between the surface and the center of the gel to be generated. Such a pH gradient varies depending on time but does not disappear even in a steady state. In contrast to the enzymatic process, the chemically induced shrinking of the gel takes place only through a pH difference between the gel (pH 4) and the outer medium (pH 6); thus, in an equilibrium state there is no pH gradient. Consequently, one may say that the position dependence of the swelling ratio which has been observed in the enzymatic process should be due to a pH gradient within the gel phase. Then it is natural to conclude that the present results cannot be explained by considering osmotic pressure arising from mobile counterions within the gel phase surrounded by the Donnan potential barrier. This conclusion is supported by many more data through comparison of them with those obtained from mathematical simulations. Discussion Modeling for Simulations. To confirm the above conclusion, we must make clear the concentration profiles for both the substrate and the product within the gel. For
this purpose, a system in which diffusion proceeds in parallel with reaction should be considered. Then, one might argue that there was a position-dependent change in diameters between the top and the bottom of the gel; in other words, our gel is not a “real” cylinder in shape. In such a case, many difficulties would arise in the mathematical description of a reaction-diffusion equation. A closer look at the gel in Figure 1 (see P-2), however, reveals that there is little difference in diameters over a considerably wide range around the center. In this range, we may neglect the diffusion of substrate molecules through the circular area at the top and at the bottom because of consumption of the substrate by enzyme reaction. Therefore, as shown in Figure 2, the central part of our gel system may be taken as an infinitely long circular cylinder of radius R. This approximation has long been used to provide a reaction-diffusion model for biological systems such as a red muscle fiber (e.g., see ref 18). Descriptions of Enzymatic Change. Prior to the mathematical expression of the diffusion process based on the above cylindrical model, we attempted to provide the rate of hydrolysis of urea with urease. This enzyme reaction has been believed to proceed via a two-step mechanism14
E + S h ES f ES′ + P1 f E + P2
(2)
where E is the enzyme, S is the substrate, ES is the enzyme-substrate complex, ES′ is the carbamate intermediate [H2NC(dO)-enz], and P1 and P2 are, respectively, ammonia and ammonium carbonate. Thus, the reaction rate (v) as a function of the concentrations of protons (H+) and substrates (S) can be given by the following equations:19
v)
kcat[E][S] KmC1 + [S]C2
(3)
(18) (a) Wyman, J. J. Biol. Chem. 1966, 10, 115. (b) Murray, J. D. J. Theor. Biol. 1974, 47, 115. (c) Britton, N. F. Reaction-Diffusion Equations and Their Applications to Biology; Academic Press: London, U.K., 1986; pp 239-247. (19) Lailder, K. J.; Bunting, P. S. The Chemical Kinetics of Enzyme Action, 2nd ed.; Clarendon Press: Oxford, U.K., 1973; pp 142-149.
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C1 ) 1 + 10a[H+] +
1 10 [H+]
(4)
C2 ) 1 + 10c[H+] +
1 10 [H+]
(5)
b
d
Here, kcat (1.50 × 103 s-1) denotes the reaction rate constant, Km (2.82 × 10-3 M) is the Michaelis constant, and the brackets signify the molar concentration of the corresponding species. Moreover, constants a-d in eqs 4 and 5 are a ) 5.92, b ) 6.86, c ) 5.38, and d )7.12. These constants as well as kcat and Km were experimentally determined by carrying out the activity measurements at pH 5.5-7.0. Numerical Simulations. Using the cylindrical model in Figure 2, the following parabolic partial differential equations are sufficient to simulate the transient variations of substrate and product concentrations in the gel phase (e.g., see ref 18);
( ) D ∂ ∂[X] ∂[X] ) 2v + r ∂t r ∂r( ∂r ) DS ∂[S] ∂[S] ) -v + r ∂t r ∂r X
(6) (7)
Here DS and DX are diffusion coefficients for the substrate and the product, respectively. Moreover, [X] is the molar concentration of the product, and from eq 1 it may be written as
[X] ) [NH4+] ∼ (1/2)[CO32-]
(8)
For the present purpose, however, we must rewrite eq 8 as a function of [H+] (therefore, pH). There are theoretical and experimental approaches for obtaining a relationship between [X] and [H+]. The former is based on both the electroneutrality condition and the acid-base equilibrium,20 whereas the latter relies on the measurements of pH and [X] at different stages of urease-catalyzed hydrolysis of urea (see Experimental Section). In this study we employed the experimental approach and rewrote eq 8 as
[X] ) 1.05 × 10-7[H+]-0.731
(9)
For eqs 6 and 7, the accompanying initial and boundary conditions are written as
[S] ) 0 and [X] ) 0 ∂[X] ∂[S] ) 0 and )0 ∂r ∂r [S] ) [S]0 and [X] ) 0
at t ) 0
(10)
at r ) 0 and t > 0 (11) at r ) R and t > 0
(12)
(20) We may theoretically obtain the following equation: [X] )
{
Kb + [H+] +
(Kb/2) + [H ]
}(
+ 10-14 Ca[H ] + 2CaK2a - [H+] - [Na+] + + [H ] [H+] + K2a
)
where Kb and K2a, respectively, denote the ionization constants of H2CO3 and CH(COOH) ) CH(COO-). Also, [Na+] is the molar concentration of sodium, and Ca is equal to [CH(COOH) ) CH(COO-)] + 2[CH(COO-) ) CH(COO-)], both values of which may be determined the concentration of the buffer solution used as the solvent. Although the difference between the theoretical and experimental results was ∼8% at maximum (at pH 4.5), we used the experimentally obtained curve of [X] vs pH because the activity coefficient of each species was neglected in the theoretical estimation.
Introducing the dimensionless variables (U ) [S]/[S]0, V ) [X]/[S]0, θ ) DXt/R2, and η ) r/R) and the dimensionless parameter (φ ) kcatR2/DX) into eqs 3, 6 and 7, we obtain
∂U 1DS∂U DS∂2U ) -v′ + + ∂θ ηDX ∂η DX ∂η2
(13)
1∂V ∂2V ∂V ) 2v′ + + ∂θ η ∂η ∂η2
(14)
φ[E]U KmC1 + U[S]0C2
(15)
v′ )
and the initial and boundary conditions for eqs 13-15 are written as
U ) 0 and V ) 0 ∂V ∂U ) 0 and )0 ∂η ∂η U ) 1 and V ) 0
at θ ) 0
(16)
at η ) 0 and θ > 0 (17) at η ) 1 and θ > 0
(18)
We solved eqs 13-18 numerically with the aid of a computer using a finite difference method. The adopted values of DS (1.65 × 10-6 cm2‚s-1) and DX (2.80 × 10-6 cm2‚s-1) were calculated from the diffusion coefficient (1 × 10-5 cm2‚s-1) of protons in the literature21 using the , where empirical relationship22 D ) 9.870 × 10-5 M-0.4404 W MW is the molecular weight of a solute. Permeability of various dialysis membranes for urea as a permeant has been studied in connection with development of hemodialyzers.23 Thus, it is also possible to estimate DS from other literature23 using a relation of DS ) Pl/S, where P denotes the permeation coefficient (cm‚s-1), l is membrane thickness (cm), and S is the partition coefficient. The value of DS [(1.62 ( 0.16) × 10-6 cm2‚s-1] obtained from the permeation experiments [P ) (3.77 ( 0.25) × 10-4 cm‚s-1; l ) 42.2 ( 9.6 µm] with ethylene-vinyl alcohol copolymer membranes having huge pores23 was in fair agreement with that from the calculation, suggesting the validity of our estimation of DS as well as DX. However, it was very difficult to learn about the effect of the network charges on the diffusion of ionic solutes such as NH4+ and CO32-, so that we neglected this effect in our simulations. With this restriction, we made simulations from which a set of distribution curves of [S] and pH against r/R was obtained. At first the simulation was made to learn how the distribution curves of [S] and pH vary with time. Figure 3 shows typical results of simulations that were performed for the gel in Figure 1a. Under conditions where swelling equilibrium has not yet been attained (i.e., times < 30 min), simulations with eqs 13-18 are really not correct. Nevertheless, it is significant to outline time-dependent changes in both substrate concentration and pH within the gel. Thus, we tentatively used the observed values of R (i.e., d2/2) at different times in the calculations of the dimensionless variables θ () DXt/R2) and η () r/R) as well as the dimensionless parameter φ () kcatR2/DX). Then, [E] was calculated as [E]0 × (d2/d0)-3, where [E]0 (0.40 mg/ (21) Ohmori, T.; Yang, R. Y. K. Biotechnol. Appl. Biochem. 1994, 7, 486. (22) Takezawa, S.; Ozawa, K.; Mimura, R.; Sakai, K. Jinnkou-Zouki (Artificial Organs) 1984, 13, 1460 (in Japanese). (23) Sakai, K.; Takesawa, S.; Mimura, R.; Ohashi, H. J. Chem. Eng. Jpn. 1987, 20, 351.
pH Gradients within Polyelectrolyte Gels
Figure 3. Simulated distribution curves of substrate concentration ([S]) and pH within the gel phase as a function of θ and φ[E]. Simulations were performed using data in Figure 1a: (a) 0.1 min; (b) 3 min; (c) 5 min; (d) 7 min; (e) 14 min; (f) 80 min.
mL) and d0 (471 µm) denote the enzyme concentration and the diameter of the gel in its preparation, respectively. From the [S] versus r/R curves, it is found that the substrate concentration within the gel phase initially increases (curves a-c), subsequently decreases (curve d), and finally reaches a steady state (curves e and f). These changes are due to the fact that diffusion proceeds in parallel with reaction. From the curves of pH versus r/R, a pH gradient between r/R ) 0.5 and 1 is observed because the resulting products diffuse into the outside through the surface. In particular, in a steady state the pH gradient observed is found to become very sharp (see curves e and f). Figure 4 shows the distribution curves of pH and substrate concentration against r/R as a function of φ[E] at φ ) 10, at which the system is in a steady state. An increase in φ[E] (therefore, an increase in the immobilized enzyme concentration) enhances the sharpness of a fall in [S] as well as a rise in pH at the range from the surface to the center. In a steady state, therefore, we may point out the following significant prospects: (i) The pH gradient within the gel disappears as the immobilized enzyme concentration approaches infinity, because under such a condition the enzyme reaction takes place only at the liquid-gel interface. (ii) The shrinking due to a broad distribution of pH, as can be seen at φ[E] < 0.1, becomes more remarkable in a gel with a large radius rather than that with a small radius; in other words, a local pHdependent change in swelling ratio amplifies with increasing R. Comparison of Simulated and Experimental Results. All of the experimental results have been obtained as time-dependent changes in gel diameter as a function of the size of cylindrical gel as well as the amount of immobilized urease. To compare these results with those obtained through the simulation, we must learn about
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Figure 4. Simulated distribution curves of substrate concentration ([S]) and pH within the gel phase as a function of φ[E] at θ ) 10. Values of φ[E] are given in each panel.
Figure 5. Experimentally obtained pH dependence of equilibrium swelling ratio (RS). Measurements were carried out at 35 °C using 5 mM maleate buffers with a variety of pH values. RS was calculated by dividing the diameter (de) at equilibrium swelling by the diameter (d0) in the preparation. Plots show the average from five gel samples with different sets of d0 (µm) and [E]0 (mg/mL): d0 ) 290 and [E]0 ) 0.2; 290 and 8.0; 471 and 0.4; 471 and 4.0; and 637 and 2.0, respectively. Standard deviation was 5% at the maximum and within limits of experimental error. Full line was then obtained by curve fitting of each plot by eq 19.
the pH dependence of the swelling ratio (RS). Equilibrium gel diameter (de) at a given pH was thus normalized by the inner diameter (d0) of a capillary used in the gel preparation and plotted against pH (see Figure 5). By use of a curve fitting method on a computer, the following polynomial equation was obtained (correlation coefficient ∼0.999);24
RS ∼ de/d0 ) 0.068(pH)3 - 1.11(pH)2 + 5.55(pH) 6.93 (19) Then, we assumed that the overall swelling ratio (〈RS〉) of
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Table 1. Comparison of Time Courses of Enzymatic Gel Shrinking Observed by Experiments and Calculated by Mathematical Simulationsa time Robsd [E] × 107 b (min) (µm) (mol/L) 1 2 3 4 5 6 7 8 10 12 14 16 20 30 40 80
458 455 453 448 440 430 423 410 400 390 383 379 370 366 365 365
1.14 1.16 1.17 1.21 1.28 1.37 1.44 1.58 1.70 1.83 1.94 2.00 2.15 2.22 2.24 2.24
∂U/∂θd ∂V/∂θd φ[E]e Rcalcd ∆f η ) 0.75 η ) 0.50 η ) 0.25 η ) 0.00 η ) 0.75 η ) 0.50 η ) 0.25 η ) 0.00 (mol/L) (µm) (µm)
θc 0.080 0.162 0.246 0.336 0.434 0.545 0.659 0.800 1.050 1.325 1.603 1.871 2.454 3.762 5.044 10.08
3.25 1.30 0.70 0.35 0.05 -0.50 -1.00 -0.30 0.03 0.01 0.00 0.00 0.00 0.00 0.00 0.00
3.10 2.05 1.30 0.70 0.05 -0.95 -1.35 -0.40 0.03 0.01 0.00 0.00 0.00 0.00 0.00 0.00
1.40 2.10 1.65 0.95 0.10 -1.15 -1.30 -0.40 0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00
0.70 2.05 1.80 1.05 0.10 -1.15 -1.25 -0.40 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00
0.25 0.25 0.30 0.35 0.60 1.25 1.60 0.25 -0.15 -0.03 0.00 0.00 0.00 0.00 0.00 0.00
0.25 0.40 0.50 0.70 1.15 2.30 2.35 0.30 -0.20 -0.03 0.00 0.00 0.00 0.00 0.00 0.00
0.15 0.45 0.65 0.90 1.55 2.90 2.50 0.30 -0.25 -0.04 0.00 0.00 0.00 0.00 0.00 0.00
0.10 0.45 0.70 1.00 1.65 3.05 2.50 0.25 -0.25 -0.04 0.00 0.00 0.00 0.00 0.00 0.00
0.127 0.128 0.129 0.130 0.132 0.135 0.138 0.142 0.145 0.149 0.152 0.154 0.157 0.159 0.159 0.159
438 437 433 426 413 384 355 342 343 344 344 344 343 343 343 343
20 18 20 22 27 46 68 68 57 46 39 35 27 23 22 22
a All of the data were obtained by use of a gel with d ) 471 µm and [E] ) 0.4 mg/mL or 8.33 × 10-7 mol/L in the preparation. b Calculated 0 0 by [E] ) [E]0/RS3, where RS is the swelling ratio. c Calculated by θ ) DXt/R2 as R ∼ Robsd. d Calculated at different values of dimensionless radius; η () r/R). e Calculated by φ[E] ) (kcat[E]R2)/DX as R ∼ Robsd. f Calculated by ∆ ) Robsd - Rcalcd.
a gel within which a pH gradient has been enzymatically formed would be calculated by
〈RS〉 )
1
gel sample
n
∑RS,i
ni)1
Table 2. Comparison of Equilibrium Swelling Ratios Obtained from Experiments and Simulations
(20)
where RS,i denotes a local swelling ratio at an arbitrary radial division i (i ) 1, 2, ..., n) and may be determined from the pH versus r/R curve using eq 19 and n is a total of radial divisions. In our simulations we divided the dimensionless radius r/R into 20 because at n > 15 there was little difference in the calculated 〈RS〉 values. First, we compared the experimental results in Figure 1 with those of simulations (Table 1). Then RS,i was approximately calculated using eq 19, which was experimentally obtained on the basis of “equilibrium” gel diameters. Correctly, this approximation is not appropriate until changes in the gel diameter stop as mentioned in the previous section. From Table 1, however, it is likely that the formation of a pH gradient within the gel phase is in a steady state (∂U/∂θ ) ∂V/∂θ ∼ 0) before reaching the swelling equilibrium (times > 30 min). This is not surprising when we consider the results obtained by Tanaka et al.25 from a study of swelling kinetics of PAAm gels in pure water. They showed that the diffusion coefficient of a gel network was 3.2 × 10-7cm2‚s-1, the value of which agreed well with that (3 × 10-7 cm2‚s-1) estimated by their laser light scattering experiments. These diffusion coefficients are ∼1/10 that of the substrate or the product, so that diffusion of the gel network should be the rate-determining step in the enzymatic shrinking process. This would be one of the reasons why there is a large difference (∆ in Table 1) between the observed and calculated radii in the time range of 5-20 min, although another reason might be due to the simulation under conditions where swelling equilibrium has not yet been attained. After the establishment of swelling equilibrium, (24) Using the results of potentiometric titration for our gels, the network charge (C in mol/g of dry polymer) as a function of pH can be given by the Henderson-Hasselbalch equation: pH ) pK + n log[(C0 - C)/C]. Here, pK (∼5.57) represents the ionization constant, n (∼1.13) is the empirical constant, which denotes the magnitude of the interaction between protons and polyion as the deviation of n from unity, and C0 (∼0.775 mol/g of dry polymer) is the overall charge of the network. Thus, we may write a relationship between RS and C as RS ∼ 0.099f(c)3 + 0.033f(c)2 - 0.56f(c) + 1.29, where f(c) ) log[(C0 - C)/C]. (25) Tanaka, T.; Fillmore, D. J. J. Chem. Phys. 1979, 70, 1214.
equilibrium swelling ratio (RS) obsdb calcdc,d
d0 (µm)
[E]0 a (mg/mL)
[E] × 107 (mol/L)
φ[E] (mol/L)
290 290 290 290 290 290 290 290 290 290
0.20 0.30 0.40 0.50 0.75 1.00 1.50 2.00 4.00 8.00
0.658 1.05 1.70 3.09 5.34 9.93 14.8 19.9 38.8 89.6
0.0253 0.0387 0.0551 0.0701 0.123 0.183 0.274 0.365 0.726 1.53
1.85 1.81 1.70 1.50 1.43 1.28 1.28 1.28 1.29 1.23
1.85 (1.85) 1.83 (1.82) 1.74 (1.54) 1.56 (1.43) 1.48 (1.35) 1.45 (1.31) 1.42 (1.27) 1.41 (1.25) 1.38 (1.22) 1.36 (1.19)
471 471 471 471 471 471 471 471 471
0.20 0.30 0.40 0.50 0.75 1.00 1.50 2.00 4.00
0.980 1.62 2.24 4.33 8.00 11.5 17.2 22.9 48.2
0.0763 0.118 0.159 0.231 0.373 0.506 0.759 1.01 2.06
1.62 1.57 1.55 1.34 1.25 1.22 1.22 1.22 1.20
1.57 (1.44) 1.49 (1.36) 1.46 (1.32) 1.43 (1.28) 1.40 (1.25) 1.39 (1.24) 1.38 (1.22) 1.37 (1.21) 1.36 (1.19)
637 637 637 637 637 637 637 637
0.20 0.30 0.40 0.50 0.75 1.00 1.50 2.00
1.26 2.14 3.17 4.74 8.00 11.2 17.6 24.1
0.152 0.237 0.328 0.436 0.679 0.917 1.40 1.88
1.49 1.43 1.38 1.30 1.25 1.23 1.21 1.20
1.46 (1.33) 1.43 (1.28) 1.41 (1.26) 1.40 (1.24) 1.38 (1.22) 1.38 (1.21) 1.37 (1.20) 1.36 (1.20)
a Calculated by [E] ) [E] /(M R 3), where M is molecular weight 0 W S W (4.8 × 105) of urease. b Denotes (d/d0). c Denotes 〈RS〉 of eq 20. d Values in parentheses were obtained by assuming D ) D () S X 2.80 × 10-6 cm2‚s-1).
however, there is a good agreement between the observed and calculated radii, that is, ∆ ∼ 22 µm, which is within the experimental error ((13 µm). As a result, it is evident that, at least after equilibrium swelling is reached, the overall swelling ratio is identical to the sum of local swelling ratios (RS,i), the values of which are uniquely dependent on a pH distribution within the gel. In Table 2, the equilibrium swelling ratios obtained by experiments using gels with different diameters (d0) as well as different amounts ([E]0) of immobilized urease in their preparations were compared with the simulated
pH Gradients within Polyelectrolyte Gels
results (〈RS〉) at θ ) 10 (i.e., in a steady state). At [E]0 < 0.5 mg/mL there are considerably good agreements between the experimental and simulated results. For the gels with a large quantity of the immobilized enzyme, however, the simulation overestimates equilibrium radius (therefore, equilibrium swelling ratio); for example, ∆ () Robsd - Rcalcd) ) -38 µm at d0 ) 471 µm and [E]0 ) 4.0 mg/mL. This ∆ value is larger than the usual experimental error ((13 µm). To look at the cause of overestimation, we considered that in our simulations the effect of the network charges on diffusion of ionic solutes has been neglected. Thus, we made recalculations of 〈RS〉 in such a way that a DS/DX ratio slightly varies. Shown in parentheses in Table 2 are the results obtained by assuming that DS ) DX ) 2.80 × 10-6 cm2‚s-1. At [E]0 > 0.75 mg/mL there is a good agreement between the experiment and the simulation, whereas for the gels with a low quantity of the immobilized enzyme the simulation underestimates equilibrium radius; for example, ∆ ) 42 µm at d0 ) 471 µm and [E]0 ) 0.2 mg/mL. Nevertheless, two significant prospects [(i) and (ii) from the simulation, which have been mentioned in the previous section] are fully confirmed by the experiments, even when DS/DX was varied slightly. In addition, prospect i was confirmed by the fact that the characteristic difference in the diameters (d1 ∼ d3 > d2) disappears when the amount of the immobilized enzyme becomes very high (i.e., [E]0 > 2.0 mg/mL). As a result, it has become apparent that the pH gradient generated within the gel phase is the chief factor from which the gel volume is uniquely determined. We could not simulate aspects of pH distributions near the top and bottom of a cylindrical gel, both parts of which have been found to undergo a slight shrinking change in a flow of a substrate solution. However, the shrinking characteristics observed at the central part have been accounted for in connection with the formation of pH gradients. Consequently, it is reasonable to consider that the shrinking changes at the top and bottom are also related to a pH gradient formed within the gel phase near both parts. Conclusions Thermally responsive cationic gels with the immobilized enzyme, in the form of a small cylinder, were prepared via physical entrapment of urease within the copolymer network consisting of NIPA and VI. The position-depend-
Langmuir, Vol. 18, No. 15, 2002 5667
ent change in the swelling ratio for a gel was observed when it was placed in a flow of a substrate solution, by which both pH and substrate concentration remained constant throughout the experimental period. To understand this observation in terms of a pH gradient that is formed within the gel phase due to the immobilized enzyme reaction, mathematical simulations using a reactiondiffusion model were carried out by taking the central part of the gel as an infinitely long circular cylinder. There was good agreement between the simulated and experimental results. From the present study, it is reasonable to mention that (i) a charge distribution depending on the pH gradient would appear in the gel and (ii) this affects its overall swelling ratio. These can no longer be explained in terms of the concept of osmotic pressure arising from mobile counterions within the gel phase. Another important piece of information from the present study is that the enzymatically generated pH gradient can be regulated by varying either the amount of immobilized enzyme or the size of gel in its preparation. This suggests a technological development in the construction of a “biochemo-mechanical system” (see refs 26 and 27) capable of converting biochemical energy created as a result of an immobilized enzyme reaction into mechanical work through the swelling and shrinking of the gel, thereby making this type of immobilized enzyme distinct from those of the more usual sort from the perspective of its utilization as a biocatalyst in a chemical conversion. Acknowledgment. This work was supported in part by a Grant-in-Aid for Scientific Research to E.K. from the Ministry of Education, Japan (No. 08558092). LA0115177 (26) For examples, see: (a) Kokufuta, E.; Zhang, Y.-Q.; Tanaka, T. Nature 1991, 351, 302. (b) Kokufuta, E. Prog. Polym. Sci. 1992, 16, 647. (c) Kokufuta, E. Adv. Polym. Sci. 1993, 110, 159. (d) Kokufuta, E. Functional Immobilized Biocatalysts Prepared Using Stimulus-sensitive Polymer Gels. In The Polymeric Materials EncyclopediasSynthesis, Properties and Applications; Salamone, J. C., Ed.; CRC Press: New York, 1996; Vol. 4, F-G, pp 2615-2621. (e) Ogawa, Y.; Ogawa, K.; Wang, B.; Kokufuta, E. Langmuir 2001, 17, 2670. (27) Biochemically induced swelling or shrinking changes observed in ref 26 have been qualitatively explained in terms of forming a product distribution within the gel phase. Our previous experiments were, however, performed in a closed system, so we could not observe the position-dependent changes of swelling ratio as reported here.
