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Effect of enzyme concentration on the dynamic behavior of a membrane-bound enzyme system
Journal of Membrane Scaence, IO (1992) 237-247 Elsevler Science Pubhshers B V , Amsterdam
237
Effect of enzyme concentration on the dynamic behavior of a membrane-bound enzyme system Naoto Sakamoto Znstrtute of Znformatlon Scrences and Electronics, Unrversrty of Tsukuba, Tsukuba-shl, Zbarakz305 (Japan) (Received April 30,1990, accepted m revised form February 6,1992)
Abstract The dynamic behavior of the reaction-diffusion system, composed of glucose ox&se (EC 1 13 4) lmmoblhzed at a uniform concentration m a membrane, used as a glucose electrode 1s represented by a diffusion equation with a nonlinear reaction-term m one-dimensional space The mathematical model 1s analyzed by computer amulatlon, that is, numerical integration of the equation under various mltml and boundary condltlons, to examme the effect of enzyme concentration on the response charactenstlcs (responsiveness and hneanty m response) of the electrode The analysis of the responses of the system to stepwlse changes m the boundary value (glucose concentration m sample solution) infers that the enzyme concentration governs the pattern of the spatial dlstnbutlons of the substrates (glucose and dissolved oxygen) m steady states and transient responses It IS also revealed that the response characterlstlcs of the electrode are optimized with concentration of nnmoblllzed enzyme and that the system establishes the steady states at the same spatial dlstnbukons of the substrates, regardless of the boundary value The dlffuslon of the substrates and the oxygen concentration also have slgmficant effects on the response charactenstlcs of the electrode Keywords
enzyme electrode, lmmoblhzed teristics, computer simulation
glucose oxldase, reactlon-dlffuslon
Introduction Membrane-bound enzyme systems perform fundamental functions in cellular metabolism and many blochemxal reactors The catalytic and regulatory activity of enzymes m a metabolic pathway is attained efficiently through then structural localization either m the membrane of a cell organelle or m the cytoplasm. Enzymes immobilized m suitable membranes Correspondence to Naoto Sakamoto, Institute of Information Sciences and Electromcs, Umverslty of Tsukuba, Tsukuba-shl, Ibarakl305, Japan
0376-7388/92/$05
00 0 1992 Elsevler Science Pubhshcn
system, response charac-
can function as sensor element in enzyme electrodes and as reacting component m bioreactors. The analysis of the reaction kinetics of such membrane-bound enzymes 1sessential for clarification of the relationship between the enzymatic reactions and the functions of the biochemical systems. The dynamic behavior of these &stnbuted reaction systems IS represented as a reaction-diffusion system (RD system) [ 1,2] The computer simulation [ 31 of an RD system, that IS,numerical integration of the nonlinear partial differential equations for the RD system (&ffuslon equations with nonlinear reactron-terms), yields the temporal and spa-
B V All rights reserved
238
tealvariations m concentrations of every chemical species in the system, leading to a quantitative description of the dynamic behavior of the system, from which the dynamic characteristics can be derived for membrane-bound enzyme systems. This article concerns the dynamic behavior of glucose oxidase (EC 1.1.3.4) immobilized at a uniform concentration in a membrane used as a glucose electrode. The RD system for this membrane-bound enzyme system is expressed by a set of diffusion equations in one-dlmenslonal space with terms expressing the reaction rates of glucose oxidase and the specific boundary conditions. The computer simulation (numerical integration of these equations under various initial and boundary conditions) is performed to analyze the effect of enzyme concentration on the dynamic behavior of the RD system The responses of the system with various enzyme concentrations to stepwise changes of various magnitudes m glucose concentration in sample solutions (i.e. the response to various changes in the boundary comhtion) are examined to reveal the temporal variation m spatial distribution of the substrate concentrations (glucose and dissolved oxygen) and reaction rate This analysis enables us to clarify the contribution of the enzymatic reaction to the responsiveness and sensitivity of an enzyme electrode and to derive the effect of enzyme concentration on the dynamic characteristics of membrane-bound enzyme systems m general. Model and methods One-dzmensronal react&on-dlffuswn model for glucose electrode In the glucose electrode as shown m Fig. 1 an RD system is considered m this study as a model for membrane-bound enzyme systems. The glucose electrode essentially comprises a membrane with immobihzed glucose oxidase m con-
N Sakamotol J Membrane Scr 70 (1992) 237-247
i lmmobAzed glucose oxldase (enzyme membrane)
sample solution
Fig 1 Reaction-diffusion
OXygt-” electrode
system m the glucose electrode
tact with the sample solution and the (oxygen ) electrode [4] The RD system thus has two boundaries: the sample-side boundary (boundary 1) separates the membrane from the sample solution and the electrode-side boundary (boundary 2) is localized between the membrane and the electrode. For a typical enzyme electrode the enzyme is uniformly distributed m a membrane such as cellulose acetate. The following processes underlie the measurement of glucose concentration by the glucose electrode. The substrates for glucose oxidase (i.e. glucose and dissolved oxygen) in the sample solution &ffuse through boundary 1 into the membrane, in which they interact with glucose oxldase and are transformed into the products, gluconate and hydrogen peroxide. The reaction of glucose oxidase (EC 1 1.3 4) virtually follows a ping-pong mechanism [ 51: kz
E + glucose “X1
- F + gluconate
k-1 (1)
F+O 2k3X2
Z E+Hg02
k--3
where E and F denote the free and reduced states of the enzyme, respectively, and X1 and X2 are the enzyme-substrate complexes The k,‘s represent the rate constants for the respective steps. In contrast to boundary 1, boundary 2 prohibits the permeation of the solutes except for a certain amount of gaseous oxygen, which is reduced by the electrode for measurement of the oxygen content at the boundary It is assumed that the oxygen concentration at boundary 2
N Sakamotoj J Membrane Scl 70 (1992) 237-247
can be measured accurately by the electrode and thus correlates to the glucose concentration at boundary 1 The probe characteristics of the oxygen electrode [ 61, which actually contribute to the function of the enzyme electrode, are not included in this model, because we are pnmarlly concerned with the dynamic characteristics of the membrane-bound enzyme system m this study In fact, it is an important advantage of the computer simulation over the empirical analysis that the behavior of the individual components of a system can be separately analyzed and integrated into the behavior of whole system The untial state of the RD system is a steady state correspondmg to a constant glucose concentration at boundary 1. The system responds to a stepwise change m glucose concentration at boundary 1, establishing a new steady state with different spatial distribution of the substrates m the system
239
of the membrane. The behavior of the reaction products is not of concern m this study and is thus not mcluded m the formulation of the model. The dynamic behavior of the RD system is thus represented by a pair of one-dimensional diffusion equations with nonlinear reactionterms for the substrates m association with rate equations (nonlinear ordinary differential equations) for the enzyme species as follows.
aA (x,t)
-= at
acw
at=
a2Akt) g ax2 -k,A (x,t)E(x,t)
D
a2ckt)
D
+
U(x,t) p= dt
ax2 -k,C(x,t)W,t)
0
k--3& (x,t)
-k,A(x,t)E(x,t)
+k,X,(x,t) Reactlon-d&won system
equattons for the RD
In mathematical modeling of the dynamic behavior of the RD system defined above the concentrations of glucose and oxygen are assumed to vary with time t and point x m a onedimensional space (0
0
L
x
Fig 2 Reaction-diffusion system m one-dnnenslonal space A (x,t) and C(x,t) express the concentrations at pomt n and time t of glucose and oxygen, respectively The value of x varies between 0 and L Glucose ox&se IS dlstrlbuted at a uniform concentration between 0 Q z
dwx,t) p= dt
a1 (x,t) dt
+k&(X,t)
k2X1 (x,t) - k,C(x,t)F(x,t)
= k,A (x,t)E(x,t)
-(k, c-=2
(2)
+k2)X1(X,t)
(x,t) =k,C(x,t)F(x,t)
dt
-
(k-3 +k4)&(x,t)
where A (x,t) and C (x,t) express the concentrations at point x and time t of glucose and oxygen, respectively, and the capital letters specifying the enzyme species m eqn. (1) also represent its concentration. The k,‘s are the rate constants as defined m eqn. (1) and the massaction law is applied for derivation of the reaction terms. D, and D, denote the diffusion constants of glucose and oxygen, respectively, m the membrane. The enzyme is immobilized
N Sakamotol J Membrane Scz 70 (1992) 237-247
at a uniform concentration membrane so that EM)
+F(@)+X1(z,t)
( =ET)
m the
+XZ!(x,t) (3)
C(O,t) =C, (constant);
ax
a3t
at x=L
(4)
At boundary 2 (i.e. x= L) the flux of oxygen to the oxygen electrode is assumed to be negligible because the flux to the electrode surface for oxygen consumption by the reduction reaction is small compared with the other characteristic fluxes m the system Stmulatton
method
The reaction-diffusion equations for the RD system, eqn (2 ) , can be solved employing the method of computer simulation [3], that is, numerical integration of the equations under the specified boundary and uutial conditions, to display the behavior of the system in response to stepwlse changes of the glucose concentration in the sample solution. The numerical mtegratlon is performed with a semldlscretlzatlon method (the method of lines coupled with the Gear method), yleldmg the temporal change m concentration distributions of every chemical species m the system In a primary application of the method of lines [7], the spatial coordmate (x-axis) is divided in a uniform grid (N equal elements) to approximate a concentration function with linear finite elements of the functions A,(t)=A(zh,t),
C,(t)=C(th,t),
where h( = L/N) 1s the mesh size of an element. The following rate equations (a system of or&nary differential equations) are thus derived from eqn. (2). U,(t) p=a,(A, dt
A (0,t) = C, (constant ) and
and ac(x’t)=O
(5)
(z= 192, ,N
for 0 < x
0
andX,,=X,(rh,t)(t)
X,,(t)=X,(zh,t),
=ET (constant)
aA (x,t) -=
E,(t)=E(rh,t),F,(t)=F(zh,t),
-2A,
dA,(t) -=ag(Ar+l dt
+C,) -k,A,E,
- 2A, +A,_l)
+k_lXI1
-klA,E, (~=2,3,...,N-1)
+k-IXI,
+k-lxlN G(t)
-2C, +C,)-k,C,F,
p=a0(C2 dt dC,(t) -=aO(CL+, dt
- 2C,+C,-1)-k~C,F,
dC,(t)
-CN+CN_-l) +
dE,(t) -= dt
(6)
(z=2,3,...,N-1)
+ LX2r ---=a,( dt
+k--3X21
-k3CNFN
k-&N
-k,A,E,+k_,X,,+k,X,, (z= 192, ,N)
y=k,X,,-k,C,F,+k_,X,, dX,,(t) ------=klA,E,dt d&(t) ---=kk3C,F, dt
(k_, +b)Xll
- (k_3 +k4)Xzz
(z=1,2,
,N)
(2=1,2,
.,N)
(z=1,2,...,N)
where a,=D,/h’ and a,=D,/h2. The stiffly stable method of Gear [ 81 is employed here for the numerical integration of eqn. (6) to yield
241
N Sakamotol J Membrane SCL 70 (1992) 237-247
the trme course of the concentration of every chemical species at the grid points. The followmg values of the parameters are used for the simulation: L= 30 pm, N= 20; lz,=lO” M-‘-set-l, k_l=3.0X103 set-‘, kz= 300 set-‘, (I&=33 n&f for glucose), k3= lo6 M-l-set-l, I~_,=150 set-‘, I~=50 set-‘, (I&=0.20 mM for oxygen), C,=2 0 mA4, C!,=O.2 mM- 10 mM, ET=0.5 mM, 1.0 mM, or 2 o mM, D,=5.0~10-~ cm2-see-’ and D, = 2.5 x 10m5cm2-see-’ Results Steady states of the RD system When the concentratrons of glucose and oxygen at boundary 1 are kept constant at C, and C,, respectively, the RD system establishes a corresponding steady state, at which the spatial distributions of the substrate concentrations m the system do not vary with time The slmulatlon for various values of C, reveals that the distrlbutlons of the glucose concentration in the system are almost identical on a relative concentration scale regardless of the boundary value On the other hand, the concentration drstributlons are significantly dependent on the enzyme concentration m the system, as shown m Fig 3. A lower enzyme concentration results in a dlstrlbution of the glucose concentration with a more shallow shape. The simulation also infers that regardless of the boundary value of glucose the spatial distribution of the oxygen concentration has slmllar shapes depending on the enzyme concentration. Obviously, the values of the drffusron coefficients of glucose and oxygen affect the dynamic behavior of the RD system The simulation with various values of D, and D, yields the expected results, 1.e. a larger coefficient causes the dlstrlbutlon to closer approach a linear concentration gradrent m the system. For the analysis of the effect of enzyme concentration on the dy-
I
20 x (grid number)
Fig 3 Spatial distnbutions of the substrate concentrations at steady states (I) ET=0 5 mkf, (II) Er=l 0 mM, and (III) ET = 2 0 n-&f The relative concentration 1s defined as the ratio absolute concentration - mmlmum concentration maximum concentration-minimum concentration On the ordinate, 0 and 10 correspond to the mimmum and the maximum concentrations, respectively The point n IS designated by grid number (unit h)
namlc behavior, D, and D, are assumed to have values comparable to those in water. Although the diffusion coefficients are dependent on the membrane structure, the analysis w&h these values for D, and D, stallleads to some dynamic charactenstics, basic and common to membrane-bound enzyme systems. Temporal behuvlor of the RD system The temporal behavior of the RD system is examined with respect to the responses of the system to stepwise changes m glucose concentration in the sample solution. Figure 4 demonstrates a typical temporal change m concentrations of glucose and oxygen at a site near boundary 1, the center of the membrane, and boundary 2 m response to an increase m glucose concentration at boundary 1 from 0.2 mM to 2 0 mM. The computer simulation for the time course of the concentration change in the system with posltlve and negative changes, of various magnitudes, m the glucose concentration indicates that a stepwise change at boundary 1 progresses toward boundary 2 with a sequential delay and a certain trail, and that a
N Sakamotol J Membrane Set 70 (1992) 237-247
242
x TIME
(set)
Fig 4 Responses of the substrate concentrations to a stepwise change of glucose concentration at boundary 1 (A) oxygen concentration, and (B) glucose concentration, at (a) n=h, (b) x=10 h, and (c) x=20 h Glucose concentration at boundary 1 increases at t = 0 02 set from 0 2 miM to2OmMmthesystemwlthE,=lOmiVf
larger change yields a steeper gradient and a longer trail m the time course The responses of the systems with various enzyme concentrations reveal that the enzyme concentration governs the responsiveness of the system (i.e. progress speed of change and time-lag of the response m the enzyme electrode). In general the system with a higher enzyme concentration responds more rapidly to a change in the boundary value, while the system with a certain optimal enzyme concentration reaches a new steady state m the least time. The dynamic behavior of the RD system responding to a change m the boundary value can be observed more clearly by looking at the temporal variation in spatial distribution of glucose and oxygen concentrations. Figure 5 shows the variation of the absolute and relative oxygen concentration m the same response of Fig. 4 The concentration starts decreasing from
(grid number)
Fig 5 Temporal vmatlon m spatial dlstnbutlon of oxygen concentration The variation m the same response of Fig 4 IS shown The numbers attached to the curves indicate the time (set) (0) 0 01 (mltlal steady state), (1) 0 025, (2) 0 06, (3) 0 1, (4) 0 4, and (5) 19 (new steady state)
1
i x (grid number)
Fig 6 Temporal vanatlon m spatial dlstrlbutlon of glucose concentration The notations are the same as m Fig 5
boundary 1 toward boundary 2 to reach a new steady-state distribution In Fig. 6 the dlstri-
243
N Sakamotol J Membrane Scl 70 (1992) 237-247
butlon of the glucose concentration is seen correspondingly to change upward with time to a new steady-state distribution. As described above, the steady states have similar distnbutlons on a relative concentration scale The simulation confirms that the distributions vary temporally m the same way for the various changes at boundary 1 and m enzyme concentration. Response chmacterlstm electrode
of the glucose
ThB effect of enzyme concentration on the dynamic behavior of the RD system is derived quantitatively from analysis of the response characteristics of the glucose electrode The responses of the electrode result from the variation of the oxygen concentration at boundary 2 m response to the change in glucose concentration m the sample solution. The response characteristics are examined with respect to linearity and time-lag of the responses of the system to stepwlse changes in glucose concentration at boundary 1. Table 1 presents the relationship of the response of oxygen concentration at boundary 2 with enzyme concentration (of 0.5 mM, 1.0 m.M and 2.0 n&f for ET) If the oxygen concentration at boundary 2 can be measured exactly by the oxygen electrode, the linearity in the response, i.e the linear relation between glucose concentration at boundary 1 and the decrease m oxygen concentration at boundary 2 from its saturated concentration, is dependent on the enzyme concentration as demonstrated m Table 1, implying that the enzyme electrode has an optimal concentration of lmmoblhzed enzyme giving the widest range in linear response. It may contribute most to this characteristic that the oxygen content m the membrane of the glucose electrode becomes a hmltmg factor for hneanty, because scarcity of oxygen as substrate seems to prevent the sys-
TABLE
1
Effect of enzyme concentration centration at boundary 2
on the response of oxygen con-
Glucose concentration at boundary 1
Oxygen concentration (% ) b at boundary 2
(mikf)” and ita decrease
(mM
for the total enzyme concentration (mM) of 10 20 05
00 02 10 20 30 40 60 80 10 0
200 (0) 196 (22) 182 (100) 165 (194) 148 (289) 131 (383) 0 973(572) 0 654(750) 0 369(906)
200 (0) 196 (21) 181 (100) 162 (200) 143 (300) 124 (400) 0 859 (600) 0 494(793) 0 171(963)
2 00 (0) 196 (20) 180 (100) 160 (200) 140 (300) 121 (395) 0 813(594) 0 424(788) 0 080(960)
“Oxygen concentration at boundary 2 at the steady state corresponding to glucose concentration at boundary 1 bThe ratlo m percent of the decrease of oxygen concentration at boundary 2 to that for 10 mM of glucose concentration at boundary 1 TABLE 2 Effect of enzyme concentration Glucose concentration at boundary 1
on the response time
Response time’ (set )
(nW
for the total enzyme concentration (mM) of 05 10 20
10 20 30 40 60 80 100
180 189 192 2 23 3 42 5 96 115
144 164 172 174 2 48 3 68 644
144 165 1 72 192 2 81 429 794
‘The response time represents the time penod between a stepwlse change m glucose concentration at boundary 1 and estabhehment of a new steady-state value for oxygen concentration at boundary 2 The change IS apphed to the system at the steady state with c,=o2mM
terns with high enzyme concentration from a linear response over a comparable range as that of the optimal system. In Table 2 the effect of the enzyme concentration on the time-lag in the response of the
N Sakamotol J Membrane SCL 70 (1992) 237-247
244
glucose electrode is shown m terms of the response time, which is defined as the time period between a stepwise change in glucose concentration at boundary 1 (from 0.2 mA4 m this case) and establishment of a new steady-state value for the oxygen concentration at boundary 2. Although a higher concentration of enzyme is expected to result in a shorter response time, the system with an enzyme concentration higher than a certain value is found to have virtually the same response time. A higher enzyme concentration rather causes a longer response time m the case of high glucose concentration at boundary 1, implymg again the existence of an optimal enzyme concentration for this response characteristic of the enzyme electrode. An mtuitive inference from the relationships indicated m Tables 1 and 2 would lead us to suppose that a high enzyme concentration attams a high reaction rate at boundary 2 and to cause a decrease m the response time as well as a deviation from the lmear response Table 3 demonstrates, however, that the glucose conTABLE 3 Effect of enzyme concentration centratIon and reactlon rate
on the responses of glucose con-
Glucose concentration at boundary 1
Glucose concentration (m)” and relatwe reactlon rate (%)b at boundary 2
(mw
for the total enzyme concentration (n&f) of 10 20 05
02 10 20 30 40 60 80 10 0
25 (2) 125 (10) 256 (20) 393 (30) 539 (41) 865 (62) 1270 (83) 1840(100)
78 40 82 127 177 294 471 884
(13) (64) (13) (20) (27) (42) (59) (72)
15 79 16 26 36 64 118 401
(05) (25) (52) (81) (11) (18) (28) (40)
“Glucose concentration at boundary 2 at the steady state correspondmg to glucose concentration at boundary 1 bThe ratlo m percent of the reactlon rate at boundary 2 to that m the system with I&.=0 5 &and C,= 10 0 mM
centration and reaction rate at boundary 2 actually are higher m the system with a lower enzyme concentration. It follows that the reaction rate is mdirectly related to the hnearity of the response and the response time. On the other hand, it should be noted m Table 3 that, if the reaction rate at boundary 2 can be measured exactly, the system with a lower enzyme concentration provides a better linear response It is thus concluded that the enzyme concentration in the membrane mherently and m an unexpected way underlies the response characteristics of the enzyme electrode. . Discussion In this study the dynamic behavior of the membrane-bound enzyme system m a glucose electrode is represented as an RD system in a one-dimensional space. By computer simulation of the mathematical model, the effect of enzyme concentration on the dynamic behavior is clarified quantitatively with respect to the responsiveness and linearity of the response of the electrode It follows from this analysis that the premises based on the properties of an enzymatic reaction m a homogeneous solution are not always valid for RD systems In fact, the system with low enzyme concentration has a narrower hnear range than the system with high enzyme concentration, which in turn cannot rapidly reach a new steady state for higher glucose concentration m the sample solution. It is evident that the detailed analysis of the dynamic behavior of the RD system is essential for elucidation of the relationship between the enzymatic reaction and the function of the enzyme electrode. The analysis can be performed effectively with the computer simulation. In the relations of the response charactenstics of the glucose electrode with the dynamic behavior of the RD system, it is of primary importance to understand how the nonlinear kinetics of the enzymatic reaction and the diffusion process of the substrates contribute to the
N Sakamotol J Membrane See 70 (1992) 237-247
linear response of the electrode. It is found in this study that the glucose electrode has an optimal concentration of immobilized enzyme which gives the widest range in linear response. More quantitative evaluation of the effects of enzyme concentration and diffusion coefficient on the dynamic behavior of RD system can be achieved with the sensitivity analysis [ 91. The dynamic and sensitivity analyses of the RD system would lead to a design of enzyme electrode assuring a linear response for the whole measuring range, since immobilization of an enzyme m a specific distribution m a membrane is feasible [lo] Another important feature in the electrode function is the time-lag in the response (i.e the time period required for reliable measurement of a concentration change in the sample solution) The effect of enzyme concentration on this characteristic is evaluated in this study in terms of the response time, which represents the net contribution of reaction and diffusion processes to the time-lag, excluding the effect of responsiveness of the oxygen electrode and the boundary behavior of the &ffusion process of glucose and oxygen. It is an advantage of computer simulation that such evaluation of net contribution to the dynamic behavior is posslble. The simulation of the responses indicates again that the glucose electrode has an optimal concentration of immobilized enzyme for the response time, although the system with a higher enzyme concentration yields a sharper change m the time course to the change m glucose concentration at boundary 1. In association with the response time, the time-lag method [ 111 can be performed by the computer simulation to clarify the effect of the enzymatic reaction on the time-lag m the diffusion process through the membrane system The time-lag is estimated to be 0.095,0.105 and 0 105 (in set) at boundary 2 for the system with ET of 0 5, 1.0 and 2.0 (in mAf), respectively, unhcating that the reaction reduces the tlme-
245
lag from 0 30 set [ = L2/ (6D,) ] down to the afore mentioned values for simple dlffusron of glucose m the membrane. The time-lag also becomes larger for the system with higher enzyme concentration, due to the lower reaction rate at boundary 2 as seen in Table 3. Nevertheless, a longer response time is observed for the system with lower enzyme concentration because the rather unstable behavior at a higher reaction rate causes the system to require a longer time to reach steady state The kinetic behavior of an enzymatic reaction m an RD system is commonly represented by an expression from the quasi-steady-state approximation of Michaehs-Menten mechanisms [ 11. In this study, on the other hand, the computer simulation is performed for the model without the approximation, revealing the unexpected effects of enzyme concentration on the response characteristics of the enzyme electrode. It is another important advantage of computer simulation that the dynamic analysis of enzyme systems is possible without employing the approximate expressions for enzymatic reaction rate Fortunately again, for homogeneous reaction systems the Michaehs equation is verified by the computer simulation to be acceptable as a vahd expression for the nonsteady-state behavior of Michaelis-Mententype reactions [ 121. The validity of the quaslsteady-state approximation should similarly be examined for Michaelis-Menten-type reactions m RD systems The result of this analysis will be described elsewhere. In this study an RD system m one-dimensional space is shown to represent effectively the basic functional characteristics of an enzyme electrode. The extension of the model to two-dimensional space makes it possible to include the shape of the electrode and the actual diffusion process m the membrane into the RD system. Of course, the analysis of RD systems is applicable to the cellular membrane systems and the analysis by computer simulation is in
N Sakamotol
246
progress at our laboratory. The computer srmulatlon of the dynamic behavior of membranebound enzyme systems will contribute more to the mathematical aspects [ 2,131 of RD systems
This work was supported m part by a Grantm-aid for scientific research (1985-1987) from the Munstry of Education, Science and Culture of Japan The author would like to dedicate the paper to the late Professor Takashr Murachi.
Km L
t
kl
Xl kt) X,,(t) x2
List of symbols
A (x,t)
A,(t) C(x,t) C,(t)
c, co 4 DO E
Ekt) E,(t) ET F
F(a) F,(t) h
concentration of glucose at point x and time t concentration of glucose at grid z (x=zh) and time t concentration of oxygen at point x and time t concentration of oxygen at grid z (x=zh) and time t concentration of glucose at boundary 1 concentration of oxygen at boundary 1 diffusion coefficient of glucose (and D,/h2=czg) diffusion coefficient of oxygen (and D,/h2=u,,) free state of glucose oxldase concentration of E at point x and time t concentration of E at grid I (x = zh) and time t total enzyme concentration reduced state of glucose oxrdase concentration of F at point x and time t concentration of F at grid z (x = zh) and time t mesh size ( =L/N) of an element m the spatial coordmate
x2
SCL 70 (1992) 237-247
rate constant (i= 1, - 1,2,3, -3,
k,
N
Acknowledgments
J Membrane
b,t)
X21(t)
4) Mlchaehs constant thickness of membrane number of elements time (set) point in one-dimensional space complex of E with glucose concentration of X1 at point x and time t concentratron of X1 at grid z (x=zh) and time t complex of F with oxygen concentratron of X2 at point x and time t concentratron of X2 at grid z (x=zh) and time t
References J -P Kernevez, Enzyme Mathematics, North-Holland, Amsterdam, 1980 N F Blltton, Reaction-Dtifuslon Equations and Then Apphcatlons to Biology, Academic Press, London, 1986 K Hayashl and N Sakamoto, Dynamic Analysis of Enzyme Systems, Spnnger-Verlag/JSSP, Berhn/Tokyo, 1986 M Koyama, Y Sato, M Alzawa and S Suzuki, Improved enzyme sensor for glucose with an ultrafiltration membrane and unmoblhzed glucose oxldase, Anal Chum Acta, 116 (1980) 307 H J Bright and D J T Porter, Flavoprotem oxldases, m P D Boyer (Ed ), The Enzymes, 3rd edn , Vol 12, Academic Press, New York, NY, 1975, p 421 V Lmek, J Smkule and V Vacek, Oxygen electrode dynamics three-layer model - chemical reaction m the liquid film, Blotechnol Bloeng ,25 (1983) 1401 M Bleterman and I Babuska, An adaptwe method of lines with error control for parabolic equations of the reactlon-dlffuaon type, J Comput Phys (63 (1986) 33 C W Gear, Numerlcal Imtlal Value Problems m Ordinary Differential Equations, Prentice-Hall, Englewood Chffs, NJ, 1971 M Koda, A H Dogru and J H Semfeld, Sensltlvlty analysis of partial differential equations with apphcatlon to reaction and diffusion processes, J Comput Phys ,30 (1976) 259
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N Sakamotoj J Membrane SCL 70 (1992) 237-247 10
11
K Nakamura, M Alzawa and 0 Mlyawakl, ElectroEnzymology/Coenzyme Regeneration, Sprmger-Verlag, Berlin, 1988 W R Vleth, Membrane Systems Analysis and Design, Hanser Publishers, Munich, 1988
12
13
N Sakamoto, Vahdlty of quasi-steady-state and transfer-function representations for input-output relation m a Mlchaehs-Menten reaction, Blotechnol Bloeng ,28 (1986) 1191 P C Fife, Mathematical Aspect of Reacting and Dlffusing Systems, Springer-Verlag, Berlin, 1979
237
Effect of enzyme concentration on the dynamic behavior of a membrane-bound enzyme system Naoto Sakamoto Znstrtute of Znformatlon Scrences and Electronics, Unrversrty of Tsukuba, Tsukuba-shl, Zbarakz305 (Japan) (Received April 30,1990, accepted m revised form February 6,1992)
Abstract The dynamic behavior of the reaction-diffusion system, composed of glucose ox&se (EC 1 13 4) lmmoblhzed at a uniform concentration m a membrane, used as a glucose electrode 1s represented by a diffusion equation with a nonlinear reaction-term m one-dimensional space The mathematical model 1s analyzed by computer amulatlon, that is, numerical integration of the equation under various mltml and boundary condltlons, to examme the effect of enzyme concentration on the response charactenstlcs (responsiveness and hneanty m response) of the electrode The analysis of the responses of the system to stepwlse changes m the boundary value (glucose concentration m sample solution) infers that the enzyme concentration governs the pattern of the spatial dlstnbutlons of the substrates (glucose and dissolved oxygen) m steady states and transient responses It IS also revealed that the response characterlstlcs of the electrode are optimized with concentration of nnmoblllzed enzyme and that the system establishes the steady states at the same spatial dlstnbukons of the substrates, regardless of the boundary value The dlffuslon of the substrates and the oxygen concentration also have slgmficant effects on the response charactenstlcs of the electrode Keywords
enzyme electrode, lmmoblhzed teristics, computer simulation
glucose oxldase, reactlon-dlffuslon
Introduction Membrane-bound enzyme systems perform fundamental functions in cellular metabolism and many blochemxal reactors The catalytic and regulatory activity of enzymes m a metabolic pathway is attained efficiently through then structural localization either m the membrane of a cell organelle or m the cytoplasm. Enzymes immobilized m suitable membranes Correspondence to Naoto Sakamoto, Institute of Information Sciences and Electromcs, Umverslty of Tsukuba, Tsukuba-shl, Ibarakl305, Japan
0376-7388/92/$05
00 0 1992 Elsevler Science Pubhshcn
system, response charac-
can function as sensor element in enzyme electrodes and as reacting component m bioreactors. The analysis of the reaction kinetics of such membrane-bound enzymes 1sessential for clarification of the relationship between the enzymatic reactions and the functions of the biochemical systems. The dynamic behavior of these &stnbuted reaction systems IS represented as a reaction-diffusion system (RD system) [ 1,2] The computer simulation [ 31 of an RD system, that IS,numerical integration of the nonlinear partial differential equations for the RD system (&ffuslon equations with nonlinear reactron-terms), yields the temporal and spa-
B V All rights reserved
238
tealvariations m concentrations of every chemical species in the system, leading to a quantitative description of the dynamic behavior of the system, from which the dynamic characteristics can be derived for membrane-bound enzyme systems. This article concerns the dynamic behavior of glucose oxidase (EC 1.1.3.4) immobilized at a uniform concentration in a membrane used as a glucose electrode. The RD system for this membrane-bound enzyme system is expressed by a set of diffusion equations in one-dlmenslonal space with terms expressing the reaction rates of glucose oxidase and the specific boundary conditions. The computer simulation (numerical integration of these equations under various initial and boundary conditions) is performed to analyze the effect of enzyme concentration on the dynamic behavior of the RD system The responses of the system with various enzyme concentrations to stepwise changes of various magnitudes m glucose concentration in sample solutions (i.e. the response to various changes in the boundary comhtion) are examined to reveal the temporal variation m spatial distribution of the substrate concentrations (glucose and dissolved oxygen) and reaction rate This analysis enables us to clarify the contribution of the enzymatic reaction to the responsiveness and sensitivity of an enzyme electrode and to derive the effect of enzyme concentration on the dynamic characteristics of membrane-bound enzyme systems m general. Model and methods One-dzmensronal react&on-dlffuswn model for glucose electrode In the glucose electrode as shown m Fig. 1 an RD system is considered m this study as a model for membrane-bound enzyme systems. The glucose electrode essentially comprises a membrane with immobihzed glucose oxidase m con-
N Sakamotol J Membrane Scr 70 (1992) 237-247
i lmmobAzed glucose oxldase (enzyme membrane)
sample solution
Fig 1 Reaction-diffusion
OXygt-” electrode
system m the glucose electrode
tact with the sample solution and the (oxygen ) electrode [4] The RD system thus has two boundaries: the sample-side boundary (boundary 1) separates the membrane from the sample solution and the electrode-side boundary (boundary 2) is localized between the membrane and the electrode. For a typical enzyme electrode the enzyme is uniformly distributed m a membrane such as cellulose acetate. The following processes underlie the measurement of glucose concentration by the glucose electrode. The substrates for glucose oxidase (i.e. glucose and dissolved oxygen) in the sample solution &ffuse through boundary 1 into the membrane, in which they interact with glucose oxldase and are transformed into the products, gluconate and hydrogen peroxide. The reaction of glucose oxidase (EC 1 1.3 4) virtually follows a ping-pong mechanism [ 51: kz
E + glucose “X1
- F + gluconate
k-1 (1)
F+O 2k3X2
Z E+Hg02
k--3
where E and F denote the free and reduced states of the enzyme, respectively, and X1 and X2 are the enzyme-substrate complexes The k,‘s represent the rate constants for the respective steps. In contrast to boundary 1, boundary 2 prohibits the permeation of the solutes except for a certain amount of gaseous oxygen, which is reduced by the electrode for measurement of the oxygen content at the boundary It is assumed that the oxygen concentration at boundary 2
N Sakamotoj J Membrane Scl 70 (1992) 237-247
can be measured accurately by the electrode and thus correlates to the glucose concentration at boundary 1 The probe characteristics of the oxygen electrode [ 61, which actually contribute to the function of the enzyme electrode, are not included in this model, because we are pnmarlly concerned with the dynamic characteristics of the membrane-bound enzyme system m this study In fact, it is an important advantage of the computer simulation over the empirical analysis that the behavior of the individual components of a system can be separately analyzed and integrated into the behavior of whole system The untial state of the RD system is a steady state correspondmg to a constant glucose concentration at boundary 1. The system responds to a stepwise change m glucose concentration at boundary 1, establishing a new steady state with different spatial distribution of the substrates m the system
239
of the membrane. The behavior of the reaction products is not of concern m this study and is thus not mcluded m the formulation of the model. The dynamic behavior of the RD system is thus represented by a pair of one-dimensional diffusion equations with nonlinear reactionterms for the substrates m association with rate equations (nonlinear ordinary differential equations) for the enzyme species as follows.
aA (x,t)
-= at
acw
at=
a2Akt) g ax2 -k,A (x,t)E(x,t)
D
a2ckt)
D
+
U(x,t) p= dt
ax2 -k,C(x,t)W,t)
0
k--3& (x,t)
-k,A(x,t)E(x,t)
+k,X,(x,t) Reactlon-d&won system
equattons for the RD
In mathematical modeling of the dynamic behavior of the RD system defined above the concentrations of glucose and oxygen are assumed to vary with time t and point x m a onedimensional space (0
0
L
x
Fig 2 Reaction-diffusion system m one-dnnenslonal space A (x,t) and C(x,t) express the concentrations at pomt n and time t of glucose and oxygen, respectively The value of x varies between 0 and L Glucose ox&se IS dlstrlbuted at a uniform concentration between 0 Q z
dwx,t) p= dt
a1 (x,t) dt
+k&(X,t)
k2X1 (x,t) - k,C(x,t)F(x,t)
= k,A (x,t)E(x,t)
-(k, c-=2
(2)
+k2)X1(X,t)
(x,t) =k,C(x,t)F(x,t)
dt
-
(k-3 +k4)&(x,t)
where A (x,t) and C (x,t) express the concentrations at point x and time t of glucose and oxygen, respectively, and the capital letters specifying the enzyme species m eqn. (1) also represent its concentration. The k,‘s are the rate constants as defined m eqn. (1) and the massaction law is applied for derivation of the reaction terms. D, and D, denote the diffusion constants of glucose and oxygen, respectively, m the membrane. The enzyme is immobilized
N Sakamotol J Membrane Scz 70 (1992) 237-247
at a uniform concentration membrane so that EM)
+F(@)+X1(z,t)
( =ET)
m the
+XZ!(x,t) (3)
C(O,t) =C, (constant);
ax
a3t
at x=L
(4)
At boundary 2 (i.e. x= L) the flux of oxygen to the oxygen electrode is assumed to be negligible because the flux to the electrode surface for oxygen consumption by the reduction reaction is small compared with the other characteristic fluxes m the system Stmulatton
method
The reaction-diffusion equations for the RD system, eqn (2 ) , can be solved employing the method of computer simulation [3], that is, numerical integration of the equations under the specified boundary and uutial conditions, to display the behavior of the system in response to stepwlse changes of the glucose concentration in the sample solution. The numerical mtegratlon is performed with a semldlscretlzatlon method (the method of lines coupled with the Gear method), yleldmg the temporal change m concentration distributions of every chemical species m the system In a primary application of the method of lines [7], the spatial coordmate (x-axis) is divided in a uniform grid (N equal elements) to approximate a concentration function with linear finite elements of the functions A,(t)=A(zh,t),
C,(t)=C(th,t),
where h( = L/N) 1s the mesh size of an element. The following rate equations (a system of or&nary differential equations) are thus derived from eqn. (2). U,(t) p=a,(A, dt
A (0,t) = C, (constant ) and
and ac(x’t)=O
(5)
(z= 192, ,N
for 0 < x
0
andX,,=X,(rh,t)(t)
X,,(t)=X,(zh,t),
=ET (constant)
aA (x,t) -=
E,(t)=E(rh,t),F,(t)=F(zh,t),
-2A,
dA,(t) -=ag(Ar+l dt
+C,) -k,A,E,
- 2A, +A,_l)
+k_lXI1
-klA,E, (~=2,3,...,N-1)
+k-IXI,
+k-lxlN G(t)
-2C, +C,)-k,C,F,
p=a0(C2 dt dC,(t) -=aO(CL+, dt
- 2C,+C,-1)-k~C,F,
dC,(t)
-CN+CN_-l) +
dE,(t) -= dt
(6)
(z=2,3,...,N-1)
+ LX2r ---=a,( dt
+k--3X21
-k3CNFN
k-&N
-k,A,E,+k_,X,,+k,X,, (z= 192, ,N)
y=k,X,,-k,C,F,+k_,X,, dX,,(t) ------=klA,E,dt d&(t) ---=kk3C,F, dt
(k_, +b)Xll
- (k_3 +k4)Xzz
(z=1,2,
,N)
(2=1,2,
.,N)
(z=1,2,...,N)
where a,=D,/h’ and a,=D,/h2. The stiffly stable method of Gear [ 81 is employed here for the numerical integration of eqn. (6) to yield
241
N Sakamotol J Membrane SCL 70 (1992) 237-247
the trme course of the concentration of every chemical species at the grid points. The followmg values of the parameters are used for the simulation: L= 30 pm, N= 20; lz,=lO” M-‘-set-l, k_l=3.0X103 set-‘, kz= 300 set-‘, (I&=33 n&f for glucose), k3= lo6 M-l-set-l, I~_,=150 set-‘, I~=50 set-‘, (I&=0.20 mM for oxygen), C,=2 0 mA4, C!,=O.2 mM- 10 mM, ET=0.5 mM, 1.0 mM, or 2 o mM, D,=5.0~10-~ cm2-see-’ and D, = 2.5 x 10m5cm2-see-’ Results Steady states of the RD system When the concentratrons of glucose and oxygen at boundary 1 are kept constant at C, and C,, respectively, the RD system establishes a corresponding steady state, at which the spatial distributions of the substrate concentrations m the system do not vary with time The slmulatlon for various values of C, reveals that the distrlbutlons of the glucose concentration in the system are almost identical on a relative concentration scale regardless of the boundary value On the other hand, the concentration drstributlons are significantly dependent on the enzyme concentration m the system, as shown m Fig 3. A lower enzyme concentration results in a dlstrlbution of the glucose concentration with a more shallow shape. The simulation also infers that regardless of the boundary value of glucose the spatial distribution of the oxygen concentration has slmllar shapes depending on the enzyme concentration. Obviously, the values of the drffusron coefficients of glucose and oxygen affect the dynamic behavior of the RD system The simulation with various values of D, and D, yields the expected results, 1.e. a larger coefficient causes the dlstrlbutlon to closer approach a linear concentration gradrent m the system. For the analysis of the effect of enzyme concentration on the dy-
I
20 x (grid number)
Fig 3 Spatial distnbutions of the substrate concentrations at steady states (I) ET=0 5 mkf, (II) Er=l 0 mM, and (III) ET = 2 0 n-&f The relative concentration 1s defined as the ratio absolute concentration - mmlmum concentration maximum concentration-minimum concentration On the ordinate, 0 and 10 correspond to the mimmum and the maximum concentrations, respectively The point n IS designated by grid number (unit h)
namlc behavior, D, and D, are assumed to have values comparable to those in water. Although the diffusion coefficients are dependent on the membrane structure, the analysis w&h these values for D, and D, stallleads to some dynamic charactenstics, basic and common to membrane-bound enzyme systems. Temporal behuvlor of the RD system The temporal behavior of the RD system is examined with respect to the responses of the system to stepwise changes m glucose concentration in the sample solution. Figure 4 demonstrates a typical temporal change m concentrations of glucose and oxygen at a site near boundary 1, the center of the membrane, and boundary 2 m response to an increase m glucose concentration at boundary 1 from 0.2 mM to 2 0 mM. The computer simulation for the time course of the concentration change in the system with posltlve and negative changes, of various magnitudes, m the glucose concentration indicates that a stepwise change at boundary 1 progresses toward boundary 2 with a sequential delay and a certain trail, and that a
N Sakamotol J Membrane Set 70 (1992) 237-247
242
x TIME
(set)
Fig 4 Responses of the substrate concentrations to a stepwise change of glucose concentration at boundary 1 (A) oxygen concentration, and (B) glucose concentration, at (a) n=h, (b) x=10 h, and (c) x=20 h Glucose concentration at boundary 1 increases at t = 0 02 set from 0 2 miM to2OmMmthesystemwlthE,=lOmiVf
larger change yields a steeper gradient and a longer trail m the time course The responses of the systems with various enzyme concentrations reveal that the enzyme concentration governs the responsiveness of the system (i.e. progress speed of change and time-lag of the response m the enzyme electrode). In general the system with a higher enzyme concentration responds more rapidly to a change in the boundary value, while the system with a certain optimal enzyme concentration reaches a new steady state m the least time. The dynamic behavior of the RD system responding to a change m the boundary value can be observed more clearly by looking at the temporal variation in spatial distribution of glucose and oxygen concentrations. Figure 5 shows the variation of the absolute and relative oxygen concentration m the same response of Fig. 4 The concentration starts decreasing from
(grid number)
Fig 5 Temporal vmatlon m spatial dlstnbutlon of oxygen concentration The variation m the same response of Fig 4 IS shown The numbers attached to the curves indicate the time (set) (0) 0 01 (mltlal steady state), (1) 0 025, (2) 0 06, (3) 0 1, (4) 0 4, and (5) 19 (new steady state)
1
i x (grid number)
Fig 6 Temporal vanatlon m spatial dlstrlbutlon of glucose concentration The notations are the same as m Fig 5
boundary 1 toward boundary 2 to reach a new steady-state distribution In Fig. 6 the dlstri-
243
N Sakamotol J Membrane Scl 70 (1992) 237-247
butlon of the glucose concentration is seen correspondingly to change upward with time to a new steady-state distribution. As described above, the steady states have similar distnbutlons on a relative concentration scale The simulation confirms that the distributions vary temporally m the same way for the various changes at boundary 1 and m enzyme concentration. Response chmacterlstm electrode
of the glucose
ThB effect of enzyme concentration on the dynamic behavior of the RD system is derived quantitatively from analysis of the response characteristics of the glucose electrode The responses of the electrode result from the variation of the oxygen concentration at boundary 2 m response to the change in glucose concentration m the sample solution. The response characteristics are examined with respect to linearity and time-lag of the responses of the system to stepwlse changes in glucose concentration at boundary 1. Table 1 presents the relationship of the response of oxygen concentration at boundary 2 with enzyme concentration (of 0.5 mM, 1.0 m.M and 2.0 n&f for ET) If the oxygen concentration at boundary 2 can be measured exactly by the oxygen electrode, the linearity in the response, i.e the linear relation between glucose concentration at boundary 1 and the decrease m oxygen concentration at boundary 2 from its saturated concentration, is dependent on the enzyme concentration as demonstrated m Table 1, implying that the enzyme electrode has an optimal concentration of lmmoblhzed enzyme giving the widest range in linear response. It may contribute most to this characteristic that the oxygen content m the membrane of the glucose electrode becomes a hmltmg factor for hneanty, because scarcity of oxygen as substrate seems to prevent the sys-
TABLE
1
Effect of enzyme concentration centration at boundary 2
on the response of oxygen con-
Glucose concentration at boundary 1
Oxygen concentration (% ) b at boundary 2
(mikf)” and ita decrease
(mM
for the total enzyme concentration (mM) of 10 20 05
00 02 10 20 30 40 60 80 10 0
200 (0) 196 (22) 182 (100) 165 (194) 148 (289) 131 (383) 0 973(572) 0 654(750) 0 369(906)
200 (0) 196 (21) 181 (100) 162 (200) 143 (300) 124 (400) 0 859 (600) 0 494(793) 0 171(963)
2 00 (0) 196 (20) 180 (100) 160 (200) 140 (300) 121 (395) 0 813(594) 0 424(788) 0 080(960)
“Oxygen concentration at boundary 2 at the steady state corresponding to glucose concentration at boundary 1 bThe ratlo m percent of the decrease of oxygen concentration at boundary 2 to that for 10 mM of glucose concentration at boundary 1 TABLE 2 Effect of enzyme concentration Glucose concentration at boundary 1
on the response time
Response time’ (set )
(nW
for the total enzyme concentration (mM) of 05 10 20
10 20 30 40 60 80 100
180 189 192 2 23 3 42 5 96 115
144 164 172 174 2 48 3 68 644
144 165 1 72 192 2 81 429 794
‘The response time represents the time penod between a stepwlse change m glucose concentration at boundary 1 and estabhehment of a new steady-state value for oxygen concentration at boundary 2 The change IS apphed to the system at the steady state with c,=o2mM
terns with high enzyme concentration from a linear response over a comparable range as that of the optimal system. In Table 2 the effect of the enzyme concentration on the time-lag in the response of the
N Sakamotol J Membrane SCL 70 (1992) 237-247
244
glucose electrode is shown m terms of the response time, which is defined as the time period between a stepwise change in glucose concentration at boundary 1 (from 0.2 mA4 m this case) and establishment of a new steady-state value for the oxygen concentration at boundary 2. Although a higher concentration of enzyme is expected to result in a shorter response time, the system with an enzyme concentration higher than a certain value is found to have virtually the same response time. A higher enzyme concentration rather causes a longer response time m the case of high glucose concentration at boundary 1, implymg again the existence of an optimal enzyme concentration for this response characteristic of the enzyme electrode. An mtuitive inference from the relationships indicated m Tables 1 and 2 would lead us to suppose that a high enzyme concentration attams a high reaction rate at boundary 2 and to cause a decrease m the response time as well as a deviation from the lmear response Table 3 demonstrates, however, that the glucose conTABLE 3 Effect of enzyme concentration centratIon and reactlon rate
on the responses of glucose con-
Glucose concentration at boundary 1
Glucose concentration (m)” and relatwe reactlon rate (%)b at boundary 2
(mw
for the total enzyme concentration (n&f) of 10 20 05
02 10 20 30 40 60 80 10 0
25 (2) 125 (10) 256 (20) 393 (30) 539 (41) 865 (62) 1270 (83) 1840(100)
78 40 82 127 177 294 471 884
(13) (64) (13) (20) (27) (42) (59) (72)
15 79 16 26 36 64 118 401
(05) (25) (52) (81) (11) (18) (28) (40)
“Glucose concentration at boundary 2 at the steady state correspondmg to glucose concentration at boundary 1 bThe ratlo m percent of the reactlon rate at boundary 2 to that m the system with I&.=0 5 &and C,= 10 0 mM
centration and reaction rate at boundary 2 actually are higher m the system with a lower enzyme concentration. It follows that the reaction rate is mdirectly related to the hnearity of the response and the response time. On the other hand, it should be noted m Table 3 that, if the reaction rate at boundary 2 can be measured exactly, the system with a lower enzyme concentration provides a better linear response It is thus concluded that the enzyme concentration in the membrane mherently and m an unexpected way underlies the response characteristics of the enzyme electrode. . Discussion In this study the dynamic behavior of the membrane-bound enzyme system m a glucose electrode is represented as an RD system in a one-dimensional space. By computer simulation of the mathematical model, the effect of enzyme concentration on the dynamic behavior is clarified quantitatively with respect to the responsiveness and linearity of the response of the electrode It follows from this analysis that the premises based on the properties of an enzymatic reaction m a homogeneous solution are not always valid for RD systems In fact, the system with low enzyme concentration has a narrower hnear range than the system with high enzyme concentration, which in turn cannot rapidly reach a new steady state for higher glucose concentration m the sample solution. It is evident that the detailed analysis of the dynamic behavior of the RD system is essential for elucidation of the relationship between the enzymatic reaction and the function of the enzyme electrode. The analysis can be performed effectively with the computer simulation. In the relations of the response charactenstics of the glucose electrode with the dynamic behavior of the RD system, it is of primary importance to understand how the nonlinear kinetics of the enzymatic reaction and the diffusion process of the substrates contribute to the
N Sakamotol J Membrane See 70 (1992) 237-247
linear response of the electrode. It is found in this study that the glucose electrode has an optimal concentration of immobilized enzyme which gives the widest range in linear response. More quantitative evaluation of the effects of enzyme concentration and diffusion coefficient on the dynamic behavior of RD system can be achieved with the sensitivity analysis [ 91. The dynamic and sensitivity analyses of the RD system would lead to a design of enzyme electrode assuring a linear response for the whole measuring range, since immobilization of an enzyme m a specific distribution m a membrane is feasible [lo] Another important feature in the electrode function is the time-lag in the response (i.e the time period required for reliable measurement of a concentration change in the sample solution) The effect of enzyme concentration on this characteristic is evaluated in this study in terms of the response time, which represents the net contribution of reaction and diffusion processes to the time-lag, excluding the effect of responsiveness of the oxygen electrode and the boundary behavior of the &ffusion process of glucose and oxygen. It is an advantage of computer simulation that such evaluation of net contribution to the dynamic behavior is posslble. The simulation of the responses indicates again that the glucose electrode has an optimal concentration of immobilized enzyme for the response time, although the system with a higher enzyme concentration yields a sharper change m the time course to the change m glucose concentration at boundary 1. In association with the response time, the time-lag method [ 111 can be performed by the computer simulation to clarify the effect of the enzymatic reaction on the time-lag m the diffusion process through the membrane system The time-lag is estimated to be 0.095,0.105 and 0 105 (in set) at boundary 2 for the system with ET of 0 5, 1.0 and 2.0 (in mAf), respectively, unhcating that the reaction reduces the tlme-
245
lag from 0 30 set [ = L2/ (6D,) ] down to the afore mentioned values for simple dlffusron of glucose m the membrane. The time-lag also becomes larger for the system with higher enzyme concentration, due to the lower reaction rate at boundary 2 as seen in Table 3. Nevertheless, a longer response time is observed for the system with lower enzyme concentration because the rather unstable behavior at a higher reaction rate causes the system to require a longer time to reach steady state The kinetic behavior of an enzymatic reaction m an RD system is commonly represented by an expression from the quasi-steady-state approximation of Michaehs-Menten mechanisms [ 11. In this study, on the other hand, the computer simulation is performed for the model without the approximation, revealing the unexpected effects of enzyme concentration on the response characteristics of the enzyme electrode. It is another important advantage of computer simulation that the dynamic analysis of enzyme systems is possible without employing the approximate expressions for enzymatic reaction rate Fortunately again, for homogeneous reaction systems the Michaehs equation is verified by the computer simulation to be acceptable as a vahd expression for the nonsteady-state behavior of Michaelis-Mententype reactions [ 121. The validity of the quaslsteady-state approximation should similarly be examined for Michaelis-Menten-type reactions m RD systems The result of this analysis will be described elsewhere. In this study an RD system m one-dimensional space is shown to represent effectively the basic functional characteristics of an enzyme electrode. The extension of the model to two-dimensional space makes it possible to include the shape of the electrode and the actual diffusion process m the membrane into the RD system. Of course, the analysis of RD systems is applicable to the cellular membrane systems and the analysis by computer simulation is in
N Sakamotol
246
progress at our laboratory. The computer srmulatlon of the dynamic behavior of membranebound enzyme systems will contribute more to the mathematical aspects [ 2,131 of RD systems
This work was supported m part by a Grantm-aid for scientific research (1985-1987) from the Munstry of Education, Science and Culture of Japan The author would like to dedicate the paper to the late Professor Takashr Murachi.
Km L
t
kl
Xl kt) X,,(t) x2
List of symbols
A (x,t)
A,(t) C(x,t) C,(t)
c, co 4 DO E
Ekt) E,(t) ET F
F(a) F,(t) h
concentration of glucose at point x and time t concentration of glucose at grid z (x=zh) and time t concentration of oxygen at point x and time t concentration of oxygen at grid z (x=zh) and time t concentration of glucose at boundary 1 concentration of oxygen at boundary 1 diffusion coefficient of glucose (and D,/h2=czg) diffusion coefficient of oxygen (and D,/h2=u,,) free state of glucose oxldase concentration of E at point x and time t concentration of E at grid I (x = zh) and time t total enzyme concentration reduced state of glucose oxrdase concentration of F at point x and time t concentration of F at grid z (x = zh) and time t mesh size ( =L/N) of an element m the spatial coordmate
x2
SCL 70 (1992) 237-247
rate constant (i= 1, - 1,2,3, -3,
k,
N
Acknowledgments
J Membrane
b,t)
X21(t)
4) Mlchaehs constant thickness of membrane number of elements time (set) point in one-dimensional space complex of E with glucose concentration of X1 at point x and time t concentratron of X1 at grid z (x=zh) and time t complex of F with oxygen concentratron of X2 at point x and time t concentratron of X2 at grid z (x=zh) and time t
References J -P Kernevez, Enzyme Mathematics, North-Holland, Amsterdam, 1980 N F Blltton, Reaction-Dtifuslon Equations and Then Apphcatlons to Biology, Academic Press, London, 1986 K Hayashl and N Sakamoto, Dynamic Analysis of Enzyme Systems, Spnnger-Verlag/JSSP, Berhn/Tokyo, 1986 M Koyama, Y Sato, M Alzawa and S Suzuki, Improved enzyme sensor for glucose with an ultrafiltration membrane and unmoblhzed glucose oxldase, Anal Chum Acta, 116 (1980) 307 H J Bright and D J T Porter, Flavoprotem oxldases, m P D Boyer (Ed ), The Enzymes, 3rd edn , Vol 12, Academic Press, New York, NY, 1975, p 421 V Lmek, J Smkule and V Vacek, Oxygen electrode dynamics three-layer model - chemical reaction m the liquid film, Blotechnol Bloeng ,25 (1983) 1401 M Bleterman and I Babuska, An adaptwe method of lines with error control for parabolic equations of the reactlon-dlffuaon type, J Comput Phys (63 (1986) 33 C W Gear, Numerlcal Imtlal Value Problems m Ordinary Differential Equations, Prentice-Hall, Englewood Chffs, NJ, 1971 M Koda, A H Dogru and J H Semfeld, Sensltlvlty analysis of partial differential equations with apphcatlon to reaction and diffusion processes, J Comput Phys ,30 (1976) 259
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K Nakamura, M Alzawa and 0 Mlyawakl, ElectroEnzymology/Coenzyme Regeneration, Sprmger-Verlag, Berlin, 1988 W R Vleth, Membrane Systems Analysis and Design, Hanser Publishers, Munich, 1988
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13
N Sakamoto, Vahdlty of quasi-steady-state and transfer-function representations for input-output relation m a Mlchaehs-Menten reaction, Blotechnol Bloeng ,28 (1986) 1191 P C Fife, Mathematical Aspect of Reacting and Dlffusing Systems, Springer-Verlag, Berlin, 1979