a simulation of active transport in an enzyme-immobilized bipolar membrane

COLLOIDS AND B SURFACES

E LS EV I ER Colloids and Surfaces B: Bioin terfaces 9 ( 1997 ) 17-29

A simulation of active transport in an enzyme-immobilized bipolar membrane

Akihiko Tanioka *, Yoshihiro Nakagawa, Keizo Miyasaka Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152,

Japan

Received 19 December 1996; accepted 13 January 1997

Abstract

In this study the transport phenomena across an enzyme-immobilized bipolar membrane was simulated, where the enzyme was fixed between positively and negatively charged layers. The transport phenomena of the products was simulated with Donnan's equilibrium and Nernst-Planck's equation of ion flux under the assumption that the hydrolytic reaction formula of urea caused by urease was represented by the following equation:

(NH2)2CO + 2H20--~ 2 N H f + C O 2 _ 3

The calculated results predicted that all products were mainly transported to the cell face to the positively charged layer, which implies an active transport in the model system. The magnitude of such a directional transport is a function of the physicochemical parameters in the membrane system, such as membrane thickness, charge density, ion mobility, substrate concentration and the concentration of the products in the membrane. The experimental results for the carbonate ion correspond well with the theoretical prediction. On the other hand, those for the ammonium ion deviate from it.

The membrane potential change generated between both sides of this membrane was calculated as a function of time. The theoretical result predicts that it has a maximum peak at the initial stage of this system. When the membrane thickness is very thin, it becomes very sharp like a pulse. The experimental results support the theoretical prediction even though it was a broad peak because the measurement was performed with a thick membrane. © 1997 Elsevier Science B.V.

Keywords." Active transport; Bipolar membrane; Donnan equilibrium; Enzyme immobilization; Nernst-Planck equation; Nonequilibrium thermodynamics; Urease

1. Introduction

Accord ing to progress made in synthetic membranes, they acquire a functionali ty o f chemical reaction in addi t ion to that o f separat ion [1]. Coupl ing between the flow and the chemical reac-

* Corresponding author. Fax: +813 5734 2876; e-mail: [email protected]

0927-7765/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PI I S0927-7765 ( 97 ) 00006-4

t ion induces the "direct ional t ranspor t" where the penetrant is carried f rom the low to the high concentra t ion side as seen in the metabolic system, which is called active t ranspor t [2]. A n enzyme- immobilized bipolar membrane is considered to be one o f the typical membranes which causes coupling between the t ranspor t and reaction [3].

The flux o f a penetrant / across a membrane is caused by the electrochemical potential difference,

18 A. Tanioka et al. / Colloids Surfaces B: Biointerfaces 9 (1997) 17-29

the fluxes of the other penetrants j 's and the chemical reaction flux in the membrane, which has been explained from the viewpoint of nonequilibrium thermodynamics as shown in the following equation [ 1 ]:

Afti Ri j Ri, J j - - - - J c h (1)

Rii j=l

where Ji is the flux of the i th penetrant, Afii the electrochemical potential of the i th penetrant, Ru the resistance to mobility of the i th penetrant mobility in the membrane, Jj the flux of the j th penetrant except i, R~ i the coupling constant, Jch the flux of the chemical reaction and R~ the coupling constant of the chemical reaction and flow. This equation suggests that no flow can be coupled with the chemical reaction in the symmet- rical membrane; R~r =0. On the other hand, they are coupled if the membrane structure is asymmet- rical; R~r ~ 0, which satisfies the condition of active transport.

A bipolar membrane, prepared in this study, has an asymmetric structure consisting of positively and negatively charged layers on opposite sides of the membrane. The enzyme is immobilized between a negative layer and a positive layer. In this system the reacted products are assumed to be 1- and 2-valent electrolytes. First, the mechanism of the directional transport of the electrolytes is theoretically investigated in this system. Nonequilibrium thermodynamics is one of the methods used as shown by Kedem et al. [3]. We, however, will approach this study differently, where the internal parameters of the membrane, such as the charge density and the ion mobility, will be adopted with the external potential, such as the concentration difference and the electrical potential difference. In addition to the theoretical treatment, these phenomena were experimentally confirmed using urease and urea as enzyme and substrate, respectively. Though the hydrolysis reaction mechanism of urea caused by urease has been unclear [5-8], in this study the following reaction formula is considered where the intermediate products are ignored, because sometimes the carbonate ion prefers to exist rather than a 1-valent anion

in the ion-exchange membrane [9].

( NH2)2CO + 2HzO~2NH ~ + CO3 2- (2)

It is suggested that this study is a general explana- tion of 1-valent anion and 2-valent cation transport across the bipolar membrane.

The bipolar membrane has interesting electrochemical properties, such as rectification and water splitting [10-15]. Kedem et al. studied the transport of the 1-valent cation and 1-valent anion across the membranes placing on electrolytic solution between the anion and cation exchange membranes [3]. Papain solution was placed into an intermediate chamber, providing N-acetyl-L-glu- tamic acid diamide from both sides of the chambers to examine the behavior of the 1-1-valent electrolytic products. They explained these phenomena using nonequilibrium thermodynamics and considered this system as a model of a biological membrane.

In this study, product concentration and membrane potential are discussed. First, transport phenomena are investigated numerically with a computer technique using the Donnan equilibrium theory and Nernst-Planck's flux equation. However, this was too complex to provide general solutions mathematically because these phenomena are carried out under nonsteady-state conditions. Therefore, it was assumed that the steady state is satisfied in this system as first approxima- tion of the simulation. Experimental systems are installed to measure the time dependence of the product concentration at both sides of the cells and the membrane potential between them.

2. Modeling

2.1. Transport phenomena

The schematic diagram of an enzyme-immobilized bipolar membrane system is shown in Fig. 1 (a). The enzyme is immobilized between positively and negatively charged layers. The interface between both charged layers is assumed to be neutral. Therefore, an enzyme-immobilized membrane is regarded as composed of 3-layers as shown in Fig. 1 (b). Transport phenomena across such a

A. Tanioka et al. / Colloids Surfaces B: Biointerfaces 9 (1997) 17-29 19

• °°

' ° ° 4

• • • °°~

(a)

• ° .

°° • °

° ° °

.) ) .)

anion and 2-valent cation, are transported from the membrane to the external solutions of sides I and/or II, across positively and/or negatively charged layers, respectively.

On simulation of this system, the following parameters are introduced: (1) concentration of substrate and products in external sides I and II, (2) partition coefficients of substrate and products in the membrane, (3) permeability coefficients of substrate and products, (4) charge densities and thickness of positively and negatively charged layers, (5) enzyme concentration at the intermediate layer, (6) enzyme kinetic constant, (7) the volume of the intermediate layer and (8) the area of the membranes.

The following assumptions are required to for- mulate this system: (1) Substrate and products f lowing across the

membrane immediately attain the steady state• (2) Electrolyte transport across the membrane can

be described by the Nernst-Planck equation, where the volume flux of water is ignored in the following treatment [ 16,17].

(3) The Donnan equilibrium is satisfied between the external solution and the membrane surface [18,19].

(4) The enzyme kinetic rate obeys the simplest Michaelis-Menten equation under the condition of simultaneous reaction [20].

(5) Electrolytes produced by the enzyme reaction are completely dissociated into their ions, which suggests that the investigation is considered as a transport phenomena of the 1-2-valent electrolyte in this case.

(b) Fig. 1. Schematic diagrams of an enzyme-immobilized bipolar membrane where+shows positively charged layer, -shows negatively charged layer and E the enzyme. (a) Single membrane model, (b) three layers model.

membrane are described as follows: Penetrant flows from external sides I and/or II into the intermediate layer through charged layers, in which hydrolytic reaction is subsequently generated by the enzyme. The products, which are 1-valent

Under the assumptions previously cited, transport equations across the membrane are estab- lished• In this study, the system is composed of a left side chamber for external solution which is denoted by subscript 0, a right side chamber denoted by subscript 2 and a bipolar membrane as shown in Fig. 2. The bipolar membrane is divided into three parts as shown in Fig. 2. These are left side part denoted by subscript L (positively charged layer), the right side part denoted by R (negatively charged layer) and the intermediate one denoted by 1 (neutral layer). Vo, V1 and V2 represent the volumes of the left chamber, neutral

20 A. Tanioka et aL / Colloids Surfaces B: Biointerfaces 9 (1997) 17-29

Vo

Cso

Cpo

PSL

PPL

IL

Vq

Cs1

Cp1

S

Ps~

PP~

[R

V2

Cs2

Cp2

L R

Fig. 2. Schematic diagram of the system with parameters and constants involved in differential equations. L indicates left side of membrane (positively charged layer), R indicates right side of membrane (negatively charged layer), 0 the left side of the chamber, 1 the neutral layer and 2 right side of chamber. Vo, VI and V2 represent volumes of neutral and chambers; Cso, Csl and Cs2 the concentration of the products; PsL and PSR are the permeability coefficients of the products; and IL and 1R are the thickness of the layers.

layer and right chamber, respectively; Cs0, Csx and Cs2 are the concentrations of the substrate in the left chamber, neutral layer and right chamber, respectively; Cp0, Cp1 and Cp2 are the concentrations of the products in the left chamber, neutral layer and right chamber, respectively, where the concentration of the 2-valent anion is one half times as much as that of a 1-valent cation; PSL and PSR are the permeability coefficients of the substrate, and PPL and PPR are the permeability coefficients of the electrolytic products across the positively and negatively charged layers, respectively. The thickness of the left and right sides of the charged membranes is expressed by IL and IR, respectively. The surface area of the membranes is noted as S. X(1)(>0) and X(2)(<0) represent charge densities of the positively and negatively charged layers, respectively. The partition coefficients of the positively and negatively charged layers, fl( 1 ) and fl(2), are substituted for approxi- mate parameters, respectively. By calculating the time dependence of the concentration of the electrolytic products in both chambers, Cpo and Cp2, the transport direction is derived. Prior to the prediction of the flow direction, the substrate and products concentration have to be estimated in the

intermediate layer and in both chambers, CS1 , Cp1 , Cpo and Cp2 from the following four differential equations, where Cs0 and Cs2 are considered constants because of their small amount

r f Cs° -- Cs1 Cs2 -- Cs1 } - - = PSL IL + PSR ~R

Ko[E]Csl 1 x s - /v~ (3) K~ + Csl

at = [-"M -r '~sl IL

Cp1 - Cp2 +PPR ~R } ×S]/V1 (4)

dCpo Cp1 - Cpo - PPL × (S/Vo) (5) dt lL

dCp2 Cp~ - C~z -- PPR × ( S / V 2 ) (6 ) dt l R

Eq. (3) gives the time dependence of the substrate concentration change at the intermediate layer. The substrate in the aqueous solution permeates through the charged membrane to the intermediate layer, subsequently being hydrolyzed to the electrolytic products to preserve the constant concentration, C1. Eq. (4) gives the time dependence of the products concentration change at the intermediate layer. The products concentration is increased by the substrate hydrolytic reaction brought about by the enzyme. They permeate through charged layers to both sides of the cells according to their concentration gradient. Eq. (5) and Eq. (6) give the time dependence of products concentration change at both sides of the cells. Because these products are electrolytes, these permeation coefficients, PPL and PPR, greatly depend on their concentration noted as Cp0, Cp1 and Cpz at each chamber. In this study, these parameters are given by solving the permeation coefficient of the 1-valent cation and the 2-valent anion across the positively and negatively charged layers. From assumptions Eqs.(2), (3) and (5), the Nernst-Planck equation was solved using the Donnan equilibrium, electrical neutral condition

ofchange:

dCsx

dt

A. Tanioka et aL / Colloids Surfaces B." Biointerfaces 9 (1997) 17-29 2 1

to derive Eq. (7) for the permeability coefficient of the cation (Teorell-Meyer-Sievers ' theory) [21,221.

P = oo+oo_ R T X F3(C; - C+)

L x

2(09+ --0)_) (o)+ +2a)_)C~ +2o9_X-] + In - - -

co+ +209_ (co+ +2co_)C~ + 2 0 ) [ X 7)

where o0_ and co+ represent anion and cation mobility, respectively, R is the gas constant, T the absolute temperature, and X the charge density of the membrane. Co, CR, CL and Co are the cation concentration at the high concentration side, membrane surfaces of the right and left sides and the low concentration side, respectively, which are calculated under the assumption of the Donnan equilibrium by the Cardano method using a computer. The amount of outflowing 2-valent cation is half times as much as that of 1-valent anion according to the condition of electroneutrality. As the simultaneous differential equations from Eqs. (3) - (6) are impossible to solve analytically because of their nonlinearity, the Runge -Kut t a - Gill method is applied for the numerical cal- culations by computer [23].

2.2. Membrane potential

The membrane potential across the enzyme- immobilized bipolar membrane was investigated as a function of time. It is thought that the membrane potential consists of the Donnan potential generated on the membrane surfaces and the diffusion potential due to the concentration gradient across a membrane. It is assumed that the Donnan equilibrium is satisfied at any time during the calculation of the Donnan potential. Substance and electrolyte flows also immediately reach the steady state, and the diffusion potential is calculated by Henderson's equation in which the concentration gradient is assumed to be proportional to the direction of the membrane thickness [24, 25]. The parameters used in this simulation and the outline of each potential including the sign are shown in Fig. 3. CLL, CLR, CRI. and CRR denote the electrolyte concentration at the membrane sur-

® Cpo ELL CLR

E2 .....

El

0 2p~ CRL C~

-3

E,

Cp2

L R

Fig. 3. Schematic diagram of the system with parameters involved in membrane potential calculation. L indicates left side of membrane (positively charged layer), R indicates right side of membrane (negatively charged layer), 0 the left side of chamber, 1 the neutral layer and 2 right side of chamber. Cp0, Cm and Cp2 are concentrations of the products; CLL, CLR, (?RE and CRR are electrolyte concentrations. El, E3, E4 and E6 are the Donnan potentials at each surface, E: and Es are diffusion potentials at each layer and AE is the membrane potential.

faces, which are derived from the simulation previously mentioned above. The Donnan potential on each surface can be described by the following equations:

R r E 1 = - - In - - (8)

F C~oql

R T + Cplq l E 3 = - In - - (9)

F C+R

R T C~L E 4 - - In - - (10)

F C~1q2

R T C~2q2 E6 = - I n - - (11)

F C+R

No potential difference is expected to be generated at the intermediate layer (AE= 0). Diffusion potentials generated in positively and negatively charged layers are described by the following two equa-

22 A. Tanioka et aL / Colloids Surfaces B." Biointerfaces 9 (1997) 17-29

tions:

E2= R T ~o+ (C~R -- C~L) + 2~o- (C~-R -- C~-L)

F co + (C{'R -- C~L) + 4co_ (C{R -- C~L)

O9+ C~R +4e~_ CLR In (12)

09+ C+L +4o~_ CL-L

R T o~ + (C~R - C;L) -~- 2CO_ (CRR -- CRL) E s -

F C0+(C~R--CffL)+4CO- C~R--C£L)

+ C~R + 4~_ C~,R In (13)

co + C~L + 4co_ C~L

Total membrane potential across an enzyme- immobilized bipolar membrane is expressed by following equation:

AE=E1 + E2 -q-E3 q-E4 + E5 -k-E6 (14)

The membrane potential transient is simulated from these equations.

3. Experimental

3.1. Membrane and experimental system

The experimental system for this study is shown in Fig. 4. The bipolar membrane which immobilized 20 mg of enzyme is clamped between a couple of glass cells. This membrane is composed of both

J (4-) ( - ) L Cell (L) Cell(R)

Fig. 4. Experimental system, where M indicates membrane, E the enzyme, L(+) left side of membrane (positively charged layer), R ( - ) right side of membrane (negatively charged layer) and S is the stirrer.

anion and cation exchange layers (Selemion CMV and AMV Asahi Glass Co.). Urease (TOYOBO Co., Urease grade II) is immobilized between these two layers. The surface area is 7.07 cm 2 and the thickness of the intermediate part of each membrane is approximately 0.005 mm. Aqueous urea solution is poured into one or both sides of the cells, and this system is regulated at 30°C. The solution in both sides is continuously stirred by magnetic stirrers.

3.2. Concentration measurements

Urea permeates through the charged layers with- out the influence of fixed charge groups in the membrane; it is subsequently hydrolyzed by the enzyme in the intermediate layer, where the hydra- tion format is shown in the introduction. The ammonium ion and the carbonate ion produced by this reaction permeate through the charged layers under the effect of their fixed charge group. Products concentration was measured every 30 min during the initial 2 h, and after that, every 1 h during the other 4 h. The concentration of the ammonium ion was determined using ion chroma- tography (Hitachi Co., Ltd.) and carbonic acid ion as the carbonate electrode (CE-235 TOA electronics, Ltd.). The carbonic acid concentration was determined using the following method: 5 ml of solution was taken out of both sides of the cell, the ionic strength adjuster was added before placing the electrode into its solution, and the voltage generating the carbonate electrode was measured.

3.3. Membrane potential measurements

For the measurement of membrane potential, the same systems are employed as previously cited. The substrate concentrations are 1M for both sides of the cells, and the membrane potential is measured using glass electrodes installed in both sides of the cells. The membrane potential change is recorded for 60 min employing the recorder. These measurements were thermostatically controlled at 20°C.

A. Tanioka et al. / Colloids Surfaces B." Biointerfaces 9 (1997) 17-29 23

4. Results and discussion

C a l c u l a t e d r e s u l t s o f p r o d u c t c o n c e n t r a t i o n

c h a n g e a r e s h o w n in F igs . 5 - 9 . T h e n u m e r i c a l

v a l u e s o f c o n s t a n t a n d in i t i a l o r b o u n d a r y condi-

tions substituted in the differential equation are shown in Appendix A. The constants and bound- ary conditions are approximated to the experimental value.

Figs. 5-7 show the time dependence of the pro-

12

3M-3M ~ ~ _ "~o_ charge density X=+0.Tmol/L

><

"" 8

c-

O

0 C.J

0 . . . . . . . . , . . . . - - - - i p 0 100 200 300 400 500

TIME ( s )

Fig. 5. Predicted time dependence of products concentration change in both sides of the cells when substrate concentrations are 3 mol 1 ~ at a constant membrane charge density of +0.7 mol 1 -~ and membrane thickness of 1 mm, respectively. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and the dotted line shows the profile faced to the negatively charged layer.

42

1M -IM charge density X =±0.7mol / I

F.

-5 28

o

103 200 300 L00 500 TIME (s)

Fig. 6. Predicted time dependence of products concentration change in both sides of the cells when the substrate concentrations are 1 mol 1-1 at a constant membrane charge density of _+0.7 mol 1-1 and membrane thickness of 1 mm, respectively. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and the dotted line shows the profile faced to the negatively charged layer.

24 A. Tanioka et al. / Colloids Surfaces B." Biointerfaces 9 (1997) 17 29

45

0.1M-O.IM x chage density X:+O.7mol/I

30

C O

t_

15 U IZ O

o

J - side

0 100 200 300 400 500

TIME (S]

Fig. 7. Predicted time dependence of products concentration change in both sides of the cells when the substrate concentrations are 0.1 mol I ~ at a constant membrane charge density of _+0.7 mol 1-1 and membrane thickness of 1 mm, respectively. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and the dotted line shows the profile faced to the negatively charged layer.

~o

x

c o

8

L5

30

15

0 0

1M-1M chorge density X=_+IO mo[/ t

: sLd#_ . . . . . . . . . . . . . . . . . . . . . . .

100 200 300 400 500

TIME (s)

Fig. 8. Predicted time dependence of products concentration change in both sides of the cells when the charge density of the cation and anion exchange layers is equal to 4-10 mol l t at a 1 mol 1-1 of substrate concentration and membrane thickness of 1 mm, respectively. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and dotted line shows the profile faced to the negatively charged layer.

ducts concen t ra t ion change in bo th sides o f the cells when the subs t ra te concen t ra t ions are 3, 1 and 0.1 mol 1-1 at + 0 . 7 mol 1 1 o f cons tan t mem- b rane charge densi ty in the an ion and ca t ion exchange layers and 1 m m o f m e m b r a n e thickness, respectively. I f the p roduc t s are ca rbona t e and

a m m o n i u m ions, the fo rmer concen t ra t ion should be ha l f the value o f the la t ter one as shown in Eq. (2) , because the e lec t roneut ra l i ty has to be ma in t a ine d dur ing ion t r a n s p o r t t h rough the charged layers. The solid line shows the p roduc t s concen t ra t ion profi le in the c ha mbe r which is faced


27

~o 1M-1M ~-\ 18 charge density X=+7×10 - 4 m o t ~ ~ '~

~ + s i d e r "

C

0 "¢"- ~ t t i t

0 100 200 300 LO0 500 TIME (s)

Fig. 9. Predicted time dependence of products concentration change in both sides of the cells when the charge density of the cation and anion exchange layers is equal to +7.0 x 10 -4 mol 1-1 at a 1 mol 1-1 of substrate concentration and membrane thickness of 1 mm, respectively. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and dotted line shows the profile faced to the negatively charged layer.

to the positively charged layer, and the dotted line shows the profile faced to the negatively charged one. When the solid line is larger than the dotted line, the products are mainly transported to the chamber faced to the positively charged layer. As time progresses, the concentration difference between both chambers is increased, which implies that this system is a model of active transport because Eq. ( 1 ) is satisfied. The lower the substrate concentration, the larger the product concentration difference between both cells. Therefore, the directional t ransport is enhanced if the substrate concentration is low.

Figs. 8 and 9 show the time dependence of products concentration change when the charge density of the anion and cation exchange layers are equal to + 1 0 m o l l -~ and 7 .0x 1 0 - 4 m o l l -1 respectively. The solid line also indicates the concentration profile to the positively charged layer, and the dotted line indicates the profile to the negatively charged one. Effective directional transport can be also obtained if the membrane charge density is high. From the theoretical point of view active t ransport in the electrolyte system is explained using the Donnan equilibrium and the

Nerns t -Planck equation according to the assumption of Teorell-Meyer-Sievers. Therefore, it appears that the membrane-induced active transport is designed by using its characteristic parameters, such as charge density, ion mobility, activity coefficient in the membrane and partition coefficients. Observing the directional t ransport as a function of the membrane thickness, the outstand- ing effect can be obtained if thin charged layers can be prepared.

Figs. 10 and 11 show the experimental results of the products concentration change when substrate concentrations are 1 and 0.1 mol 1 -1 respectively. Open circles indicate the ammonium ion concentration change in the cell faced to the positively charged layer, and the solid circles indicate the change to the negatively charged layer as a function of time. Open triangles show carbonate ion concentration change in the cell against the positively charged layer, while the solid triangles show the change to the negatively charged layer as a function of time. In Fig. 11, the products concentration is very dilute so that its measurement is not necessar- ily accurate. Then, total amount of ammonium ion concentration is nearly equal to twice that of

26 A. Tanioka et al. Colloids Surfaces B: Biointerfaces 9 (1997) 17-29

/. i ,

_s ide(NH + i % 1M-1M

" side x 3 .'~ +(NH~)

E ~ 2

~ I ~ . ~ ~ +side {CO~-)

o 0 ~ _ ~ ' ~ " - - -T ~ , ,

0 1 2 3 4 5 6 7

TINE (hrs)

Fig. 10. Experimental time dependence of products concentration change in both sides of the cells when both sides of substrate concentration are 1 mol 1 - 1. Circles show the concentration change in ammonium ion, and the triangle shows the concentration change in carbonate ion. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and the dotted line shows the profile faced to the negatively charged layer.

I I I I I I

-side(NH~ +) /@ 0.1H- 0.1H /

x ~ / / / +side(NH~ )

c- .0.9.

~ I

. --:-.,1{-jt- o , s,

0 1 2 3 4 5 6 7

TIME (hrs)

Fig. 11. Experimental time dependence of products concentration change in both sides of the cells when both sides of substrate concentration are 0.1 mol 1-1. Circles show the concentration change in ammonium ion, and the triangle shows the concentration change in carbonate ion. Solid line shows the products concentration profile in the chamber which is faced to the positively charged layer, and the dotted line shows the profile faced to the negatively charged layer.

the carbonate concentration, which implies the validity of the assumption of urea reaction by urease as shown in Eq. (2).

Directional transport seems to be realized, because the transport phenomena of the carbonate ion corresponded with the theoretical prediction. On the other hand, the ammonium ion concentration against the negatively charged layer side is higher than that against the positively charged one, which contradicts with the theoretical results. There are several explanations for this. First, one of the most reasonable explanations is based on the fact that the products are weak electrolytes. If the products are composed of 1- and 2-valent strong electrolytes, the concentration ratio among the transported 1-valent cation and 2-valent anion is strictly 2:1 in both sides of a bipolar membrane. However, the amount of carbonate ion is more than one half times that of the ammonium ion in the cell faced to the positively charged layer. If the solution is alkaline, almost all of the carbonate ions are considered to be shifted to bicarbonate ions which are 1-valent anions. In such a case, the problem is that of 1-1-valent electrolyte permeation across a charged membrane. Therefore, the dissociation equilibrium should be considered in this transport system as shown in the following equations [26]:

2H + + C O 2 - ~ H + + HCO3 ~-H2CO 3 (15)

O H - + N H + ~ N H 3 (16)

We may expect that all kinds of ion and the neutral solute permeate across the membrane at the same time. This is a problem of solute transport in a multicomponent ion system. In these cases, three additional parameters have to be provided to solve the Teorell-Meyer-Sievers theory, which are the equilibrium constants in the charged membrane as seen in Eqs. (15) and (16). Unfortunately, it is impossible to predict these. Though the calculation was carried out under the assumption that the equilibrium constants in the membrane are the same as those in the aqueous solution, the theoretical prediction could not satisfactorily explain the experimental results. In Eq. (2) the hydrolysis reaction of urea by urease was assumed to produce


only ammonium and carbonate ions. In this study the production of carbamate ion is ignored [8]. If the carbamate ion is included in Eqs. (15) and (16), the additional equilibrium constant should be considered in the Teorell-Meyer-Sievers theory. It, however, leads to noticeable confusion in our treatment because of the increase in unknown parameters.

The membrane potential change during the enzyme reaction is calculated as a function of time using the same parameters as shown in Appendix A and in Figs. 12 and 13 when the substrate concentration of both sides of the cells is 1 mol 1 - 1, and the membrane thicknesses are 1 mm and 0.01 mm, respectively. The membrane potential has a peak at the initial stage of this reaction, and after that it decreases. The potential peak of the thin membrane is remarkably sharp compared with the thick one. Fig. 14 shows the experimental results of the membrane potential change as a function of time when the membrane thickness is 0 .4mm. The reference electrode is installed in the cell faced to the anion-exchange layer. The potential increases at the initial stage to show only one peak, and after that gradually decreases. The entire profile of the experimental result shows a figure similar to the calculated result.

We have investigated which factors can control

the transport direction and products concentration in active transport for the system of an enzyme- immobilized membrane. Almost all parameters can be varied, e.g. enzyme reaction rate, amount of enzyme, substrate concentration, produced ion concentration, membrane thickness, charge density, permeation coefficients of substrate and mobility of produced ions. From these results, we may conclude that the directional transport strongly depends on the concentration of the produced ions in the intermediate layer. The high directivity is obtained in the case of low-ionic concentration which corresponds to the high- charge density according to the Teorell-Meyer Sievers theory. These phenomena are due to the effect of the Donnan equilibrium at each part of the charged layer surface. Because the diffusion coefficients of the ions and substrate are estimated to be low in comparison with the enzyme reaction rate, the membrane process is the diffusion-deter- mining step. Therefore, the previously described view is correct.

As shown in Eq. (2), urease is decomposed into a univalent cation and bivalent anion. It has been proven that these are transported to each side of the cells separated by a bipolar membrane in different concentrations. However, it is thought that the carbonate ion is immediately transformed

E hi

900

600

300

tL = [R = l m m

100 200 300 400 500

TIME(s)

Fig. 12. Predic ted t ime dependence of m e m b r a n e po ten t i a l change (AE) when subs t ra te concen t ra t ions of bo th sides of the cells are + 0.7 mol 1 and m e m b r a n e th ickness of 1 mm, respectively. 1 mo l 1 - 1 a t a cons tan t m e m b r a n e charge dens i ty of _ 1

28 A. Tanioka et al. / Colloids Surfaces B. Biointerfaces 9 (1997) 17-29

300

900

600

2O >

E W <3

[L = [R =0.01 mm

E~ i , i

' 0 ' ' 0 0 ~00 2 0 300 400 500

TIME(S)

Fig. 13. Predicted time dependence of membrane potential change (AE) when substrate concentrations of both sides of the cells are 1 mol 1 -~ at a constant membrane charge density of +0.7 mol 1-1 and membrane thickness o f 0.01 mm.

10

I Oo 15 20

i I

5 10 T I M E ( m i n )

Fig. 14. Experimental time dependence of membrane potential change (AE) when substrate concentrations of both sides of the cells are 1 mol 1 - '

into bicarbonate ion in water. Therefore, if the products are only univalent cation and anion, it cannot be expected that the directional flow of the products will be observed in such a system. In the

electrodialysis experiment, it is observed that the bicarbonate ion is converted to the carbonate ion in the ion-exchange membrane, which supports our theoretical prediction.[4]


Acknowledgment

This work is performed using a grant aid from the Ministry of Education (63550660).

Appendix A

Table 1 Constants and initial value for simulation

V0 400.0 cm 3 /I1 0.1 cm 3

V 2 400 cm 3 PSL 3.1 x 10-3 cm-2 s-1

PSR 3.1X 10-3 cm-2 s-X

IL 0.1 cm

IR 0.1 c m

S 7.0 cm 2 K M 3.9 x 10-6moll -1 K o 45 U mg- l

1.0x 10-s moll -1 q 0.9 co+ 7.62 x 10 -3 cm 2 V- i s -1 co_ 7.18 x 10-3 cm2 V - i s -1 T 303 K R 8.3 J tool -1 E 20 mg

Cell volume of left side Volume of membrane intermediate layer Cell volume of right side Substrate permeability coefficient through positively charged layer Substrate permeability coefficient through negatively charged layer Thickness of positively charged layer (left side) Thickness of negatively charged layer (right side) Membrane area Michaelis constant Enzyme activity Initial proton concentration Partition coefficient Cation mobility in water Anion mobility in water Absolute temperature Gas constant Amount of enzyme

[2] S.G. Shultz, Basic Principles of Membrane Transport, Cambridge Univ. Press, Cambridge, UK, 1980.

[3] R. Blumenthal, S.R. Caplan and O. Kedem, Biophys. J., 7 (1967) 735.

[4] Y. Yokoyama, A. Tanioka and K. Miyasaka, J. Membr. Sci., 38 (1988) 223.

[5] E. Mack and D.S. Villers, J. Am. Chem. Soc., 45 (1923) 505.

[6] J.H. Wang and D.A. Tarr, J. Am. Chem. Soc., 77 (1955) 6205.

[7] G. Gorin, Biochim. Biophys. Acta, 34 (1959) 268. [8] K. Kobashi, Seikagaku, 44 (1972) 187. [9] M. Seno, M. Abe and T. Suzuki, Ion Exchange, Kodansha

Scientific, Tokyo, 1991. A. Mauro, Biophys. J., 2 (1962) 179. I.C. Bassignana, and H. Reiss, J. Membr. Sci., 15 (1983) 27. R. Simons, Nature, 280 (1979) 824. R. Simons, J. Membr. Sci., 82 (1993) 65. S. Mafe, J.A. Manzanares and P. Ramirez, Physical Review A, 42 (1990) 6245. P. Ramirez, H.J. Rapp, S. Reichle, H. Strathmann and S. Mafe, J. Appl. Phys., 72 (1992) 259. M. Planck, Physik. Chem., 39 (1890) 161. M. Planck, Ann. Physik. Chem., 40 (1890) 561. F.G. Donnan, Z. Electrochem., 17 (1911) 572. F.G. Donnan, Z. Physik. Chem., 68 (1934) 369. A. Fersht, Enzyme Structure and Mechanism, Freeman, San Francisco, 1977. T. Teorell, Prog. Biophys. Biophys. Chem., 3 (1953) 305. K.M. Meyer and J.F. Sievers, Helv. Chem. Acta., 19 (1936) 649. M. Higa, A. Tanioka and K. Miyasaka, J. Membr. Sci., 37 (1988) 251. P. Henderson, Z. Physik. Chem., 59 (1907) 118. P. Henderson, Z. Physik. Chem., 63 (1908) 325. H. Freiser and Q. Fernando, Ionic Equilibria in Analytical Chemistry, Wiley, New York, 1963.

[10] [111

[12] [13] [14]

[15]

[161 [17] [18] [19] [20]

[211 [22]

[23]

[24] [251 [26]

References

[1] A. Katchalsky and P. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard Univ. Press, Cambridge, MA, 1965.

a simulation of active transport in an enzyme-immobilized bipolar membrane

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