J. Phys. Chem. 1083, 87, 1235-1241
which will be present in a lipid bilayer at much higher concentrations than OH-, may be the active agent: HO-MgOEP+ H
+ H,O
= HO-MgOEP
+ H,O+
Although the equilibrium constants which we have determined for these reactions (see Table I) seem, at first glance, too small to effect the phenomenon observed in ref 1a t pH's between 5 and 7, it must be remembered that
1235
the low value of the dielectric constant within a bilayer lipid membrane will highly favor deprotonation. Acknowledgment. This work was supported by the Division of Chemical Sciences, United States Department of Energy, Washington, D.C., under Contract No. DEAC02-76-CH00016. Registry No. P+, 34606-97-8; POH, 84602-23-3;OH-, 1428030-9; H+, 12408-02-5;p-DNB-, 34505-33-4.
Theory of the Biocatalytic Membrane Electrode H. F. Hameka' h p a r t m n t of ChemlStfy, Unlverslty of Pennsylvania, Philadelphia, Pennsylvania 19 104
and G. A. Rechnltz Department of Chemistry, Unlverslty of Delaware, Newark, Delaware 19711 (Received: August 16, 1982; I n Final Form: November 3, 1982)
We present a theoretical interpretation of the mechanism of a biocatalytic membrane electrode with particular emphasis on the time dependence of the approach to steady state. We make use of a model based on diffusion from a bulk solution with diffusion constant DII into an enzyme layer of thickness L with diffusion constant DI, combined with a chemical reaction obeying Michaelis-Menten kinetics within the enzyme layer. We report an exact calculation of the time dependence of the product concentration within the enzyme layer for the high-concentrationcase and an approximate treatment for the low-concentrationcase. We find that in all cases the difference between the product concentration and its steady-statevalue is proportional to the inverse square root of the time. The approach to steady state is strongly affected by stirring and by the thickness L of the enzyme layer; it is much less dependent on the value of the diffusion constant DI.
I. Introduction An important new device for medicoanalytical and diagnostic purposes is the immobilized enzyme electrode or the biocatalytic membrane electrode. Many enzymes react specifically with one and only one organic or biological substance. If such an enzyme is trapped in a matrix layer of a thickness of the order of 0.01 cm and if the layer is in contact with a substrate solution of the substance that the enzyme reacts with, then the immobilized enzyme electrode can be used for rapid quantitative analysis of the substrate solution. Typical examples of such enzyme electrodes are the glucose electrode, described by Clark and Lyons1 and constructed by Updike and Hicks,* and the urea electrode, developed by Guilbault and M ~ n t a l v o .In ~ recently proposed electrodes the immobilized enzyme layer has been replaced by bacterial, mitochondrial, or even tissue layer^.^ The general theoretical description of the various types of biocatalytic membrane electrodes is based on the onedimensional model that we have sketched in Figure 1. Here it is assumed that the biocatalyst is distributed homogeneously in the layer 0 I X I L; we denote the enzyme layer by the Roman numeral I. The region X > L , which we denote by 11, contains the substrate solution.
Recently we derived the stationary-state solutions of the substrate and product concentration profiles for the above theoretical modeL5 We obtained analytical expressions for the concentration profiles and we al.so derived power-series expansions that could be used for convenient numerical evaluation. We should also mention the work by Morf,6 who approached the same problem from a differen t viewpoint. In the present paper we want to derive the time dependence of the approach to steady state; that is, we attempt to calculate the time-dependent solutions of the concentration profiles corresponding to specific initial conditions. We shall see that we are able to derive these time-dependent solutions only for certain limiting situations corresponding to either large or small concentrations. Nevertheless, we hope that our theoretical results contribute toward a general understanding of the time dependence of the approach to steady state. In most biocatalytic membrane electrodes the substrate solution is stirred and this aspect of the electrode must be incorporated in the theoretical description. Fortunately, the effect of stirring has been studied experimentally by Morf, Lindner, and Simon.7 Even though this study deals with neutral carriers rather than with biocatalytic mem-
~
(1) L. C. Clark and C. Lyons,Ann. N.
Y. Acad. Sci., 102,29 (1962).
(2)S.J. Updike and G . P. Hicks, Nature (London), 214,986 (1967). (3) G.G.Guilbault and J. Montalvo, Jr., J.Am. Chem. SOC., 91,2164 (1969);92,2533 (1970). (4)M. A. Arnold and G . A. Rechnitz, Anal. Chem., 52, 1170 (1980).
( 5 ) H. F. Hameka and G. A. Rechnitz, Anal. Chem., 53, 1586 (1981). (6)W.E. Morf, Mikrochim. Acta, 317 (1980). (7) (a) W.E. Morf, E. Lindner, and W. Simon, Anal. Chem., 47, 1596 (1975); (b) W.E. Morf and W. Simon in "Ion-Selective Electrodes in Analytical Chemistry", H. Freiser, Ed., Plenum Press, New York, 1978.
0022-385418312087-1235$01.50/0 0 1983 American Chemical Society
1236
The Journal of Physical Chemistry, Vol. 87,No. 7, 1983
IiD,)
’
enzyme
conc = 0
Hameka and Rechnitz
IIID,,I substrate
I x:
conc = c
11. Theoretical Model We use a theoretical model that was proposed by Blaedel, Kissel, and Boguslaskis and that we sketched in Figure 1. Here it is assumed that the enzyme (or other biocatalyst) is distributed homogeneously in the region 0 5 X S L, which we denote by I, and that the substrate solution is in the region L < X, which we denote by 11. Initially, at time t = 0, the substrate solution contains a constant concentration of reactant molecules S in region 11and no molecules S in the enzyme layer I. Subsequently, at times t > 0 the molecules S diffuse into the enzyme layer and they react with the enzyme. The reaction is described by the equation k,
E +SS E S
-+ k2
E
P
(1)
a*
= Dt/DIIs
DIp/DIIp
ap2 = CY
brane electrodes, it is useful for comparisons, especially since Morf, Lindner, and Simon’ also present a theoretical interpretation of all their data. We will return to this question when we present our theoretical results in more detail.
k-l
and the new parameters /3 = DIs/L2 PP = DIp/L2
I
L Figure 1. Theoretical model of an enzyme electrode. Region I is the enzyme layer and region I1 is the bulk solution. At time t = 0 the substrate concentration Is zero in region I and it is equal to a constant c in region 11. XZO
In order to solve the above rate equations, it is convenient to introduce the new variables x = X/L C, = Cs/KM cP = Cp/KM
= k,t/KM
The rate equations 3 and 4 are then reduced to ac, a%, W, -at = P axy - x
and to ac,
p azc,
at
a’
-=--
acp = -P-p a2cp at ap2 ax2
ax2 l e x
The initial conditions at t = 0 are cs=o O S X S l c,=c l < x cp = 0 The diffusion in the bulk solution I1 is always faster than in the enzyme layer I and the possible range of values of the parameters a and ap is given by OSaS1
OSapSl
(11)
In most enzyme electrodes the substrate solution is stirred. In our theoretical model we assume that the effect of stirring is represented by an increase in the value of the diffusion constant DII or by a decrease in the value of the parameters a. However, we do not believe that a perfect stirrer exists and this means that u always remains finite. Our assumption is confirmed by the experimental results of Morf, Lindner, and Simon.’ Here E is the concentration of the total amount of enzyme In the following section we summarize the previously in region I and KMis the Michaelis-Menten c ~ n s t a n t . ~ obtained results for the stationary states which are derived The total rate of change of the concentrations Cs or Cp by setting the time derivatives in eq 8 and 9 equal to zero. in the enzyme layer I is now represented as a sum of a In subsequent sections we attempt to derive the time-dechemical term, given by eq 2, and of a diffusion term with pendent solutions that describe the time dependence of diffusion constants DIs or DIP,namely, as the approach to steady state. Our derivations are based a2cs kZEcs on the Laplace transform method that is discussed by aCS -- -DISZ Carslaw and JaegerlO and by Crank.” at KM + cS 111. Steady State a2cp kzEcs aCP - = DIP= OIXSL (3) The steady-state solutions of the rate equations 8 are at KM -k cS known, but we will briefly recall them. Here and in the following we assume that the distribution coefficient of In region I1 the rate of change of the concentration Cs substrates is unity. or Cp is determined by diffusion only since the enzyme The steady-state solutions in region I1 (1< x ) are quite concentration E is zero here. We denote the diffusion simple. The solution of eq 9 that is compatible with the constants by DIIs or DIIpand the rate equations are initial condition 10 and that remains finite at infinity is aCs/at = DI~(a2Cs/aX2) c, = c aCp/at = DIIP(a2Cp/aX2) L < x (4) cp=o l
