1586
Anal. Chem. 1981,
53, 1586-1590
LITERATURE CITED
1.2
(1) (2) (3) (4) (5) (6) (7) (8) (9) (IO) (11) (12)
1.0
0.8 I
z
(0
z
0.6
I
W >
0.d
I
I-
a
J E
0.2
(13) (14) 115)
0.0 -0.2 -3nn
---
-isn .--
o 50 SFlfiPLE NUMBER
- i n._ n -50 ..
ion
150
200
Figure 4. Deformation of a Gaussian line (high peak). Filter degree 2M = 4. N/fwhm = 1.74, 3 (medium peaks), and 6 (low peak).
in the spectrogram, the peaks will be flattend (22,23)and the background will remain for subtraction provided that the background can be approximated by a polynomial Of degree 2M + 1 within the filter width, and the number of peaks is relatively small. For “straightening through smoothing” applications the recursive representation of Savitzky-Golay smoothing fiiters (see ref 11) can most advantageouslybe used.
(18) ( 17) (18) (19) (20) (21) (22) (23) (24)
Savitzky, A.; W a y , M. J. E. Anal. Chem. 1964, 36, 1627-1638. Ernst, R. R. Adv. Magn. Reson. 1988, 2, 1-135. Wlilson, P. D.; Edwards, T. H. Appl. Specfrosc. Rev. 1978, 72, 1-81. Madden, H. H. Anal. Chem. 1978, 50, 1363-1380. Casaletto, J. J.; Rice, J. R. Appl. Anal. 1978, 6 , 143-152. Enke, C. G.; Nieman, T. A. Anal. Chem. 1978, 46, 705A-712A. Clarke, F. J. J. IEEConf. Pub/. 1973, No. 103, 136-140. Bromba, M. U. A., Dlplomarbelt, Paderborn, 1976. Borgan, 0. Scand. AcfuarlalJ. 1979, 83-105. Grevllle, T. N. E. SIAM J. Math. Anal. 1974, 5 , 376-398. Bromba, M. U. A.; Zlegler, H. Anal. Chem. 1979, 51, 1760-1762. Tomlnaga, H.; DoJyo, M.; Tanaka, M. Nucl. Instrum. Methods 1972, 96, 69-76. Betty, K. R.: Horllck, G. Anal. Chem. 1977, 49, 351-352. Horllck, G. Anal. Chem. 1972, 44, 943-947. De Elasl. M.: Glannelll. G.: PaDoff. . . P.: Rotunno. T. Ann. Ch/m. (Rome) 1975, 65, 183-190. Zlealer, H. ADD/. Soecfrosc. 1981. 35. 88-92. Koopmans, L.‘ H. “The Spectral Analysis of Time Series”; Academic Press: New York, 1974; Example 8.9. Bromba, M. U. A.; Zlegler, H. €/ectron Lett. 1980, 16, 905-906. Hamming, R. W. “Digital Filters”; Prentlce-Hall: Englewood Cliffs, NJ, 1977. Glannelll, G.; Altamura, 0. Rev. Scl. Insfrum. 1978, 47, 32-36. Papoulls, A. I€E Trans. Inf. Theory 1977, IT-23, 631-633. Carbonneau, R.; Boiduc, E.; Marmet, P. Can. J. Phys. 1973, 51, 505-509. Marchand, P.; Velllette, P. Can. J. Phys. 1976, 54, 1309-1312. Proctor, A,; Sherwood, P. M. A. Anal. Chem. 1980, 52, 2315-2321.
RECEIVED for review September 22,1980. Resubmitted April 6, 1981. Accepted May 15, 1981.
Steady-State Theory of Biocatalytic Membrane Electrodes H. F. Hameka‘ Department of Chemistry, University of Pennsylvanla, Philadelphia, Pennsylvania 19 104
G. A. Rechnltz” Department of Chemlstty, Universm of Delaware, Newark, Delaware 1971 1
A mathematical derivation is presented for the exact solution of partlai differential equations descrlblng the steady-state response of blocafalytlc potentlometrlc membrane electrodes. The derivatlon Is based upon a previously proposed model in which the steady-state response is seen to result from a comblnatlon of dlffusion and Michaelis-Menten klnetic steps. The expressions derived In thls work yield further inslght into the behavlor of blocatalytlc potentlometrlc membrane electrodes and permlt convenlent evaluation of key parameters as a function of the malor experimental variables.
tentiometric enzyme electrodes but were limited to numerical evaluations of the resulting differential equation. We now develop an exact solution which can be more conveniently employed without the need for extensive computing facilities. If we mume, as did Brady and Carr, that the net equation governing the rate of change of substrate concentration within any portion of the biocatalyst-containing membrane surrounding the sensor must contain a diffusional mass transport term and a Michaelis-Menten kinetic term, we obtain
dC, = Doa2c,- kz[EoIC, -
ax2
at
The empirical development of bioselective membrane electrodes (I) has proceeded rapidly in recent years with the recognition that bacterial cells, mitochondria, or intact animal and plant tissue slices can (2) be employed as biocatalysta at membrane electrode surfaces in a manner analogous to conventional enzyme electrodes. The concomitant development of theoretical models and formulations has not kept pace with these practical advances; recently, Brady and Carr (3) gave a theoretical treatment of the steady-state response of POCurrent address: Institute for Physical Chemistry 111, Technische Hochschule, D-6100Darmstadt, Germany. 0003-2700/81/0353-1586$01.25/0
KM+
CB
(1)
where D,is the substrate diffusion coefficient in the membrane, C, is the substrate concentration, kz[&] is the biocatalyst activity, and KM is the Michaelis-Menten constant. It is assumed that biocatalytic activity can be treated in a manner analogous to enzyme activity for the purposes of this model. The schematic model of Brady and Carr is adapted to our notation in Figure 1 where the boundary condition is that aC,/aX = 0 at X = 0 and dC,/dt = 0 at steady state. Equation 1 then becomes
d2C,
D,-
ax2
-
0 1981 Amerlcan Chemical Society
~2[EolC,
KM + c,
=O
ANALYTICAL CHEMISTRY, VOL. 53 YO. 11, SEPTEMBER 1981
d2p
d2X/dp2
dx2
(dx/dpI3
-=-
Substituting into eq 6 and rearranging then gives
Bulk Solution
Sensor
1587
This equation can be solved exactly. We define v = dx/dp, so that
Figure 1. Schematlc dlagram of biocatalytic membrane electrode model. C, and C, refer to substrate and product concentratlons,
and
respectively.
At this point we depart from the appiroach of Brady and Carr by introducing new variables. First, we substitute CO/KM = p oir C, = pKM and X/L = x
to obtain Ds
CK"
2--d2p
- kZ[EO]KMp
dX2
KM+PKM
=o
(3)
or a2p P - - -MEolL2 -
---- 0 (4) ax2 D,KM 1 + ~1 The quantity k2[Eo]L2/D,K~can be called a,and eq 4 becomes
Equation 16 can be integrated
or Second, we adopt the notation (where Cp refers to product concentration and the diffusion coefficients of product and substrate are assumed it0 be equal) that a=-
Cs KM
(at sentior)
CP b = - (at sensor) KM
c0
- (bulk) KM It is clear that a + b = c and that a and c correspond to the quantity p a t the sensor and in the bulk, respectively. The boundary conditions, disregarding time dependence at steady state, still remains that (dp/dx),,,, = 0. Equation 5 can be rearranged to c=
d2p
ap =(6) dX2 1 + ru which is a rather awkward nonlinear differential equation difficult to solve in ternis of p as a function of x . However, since eq 6 contains p on both sides, it can bo transformed into an equation for x as a function of p. Because of reciprocity, if p is a single-valued function of x in the interval 0 lx I 1,then x is a single-valuedfunction of g in the interval
1 + 01 2v2 where p is the integration constant. This constant can be determined from the boundary conditions that dpldx = 0 at x = 0 and p = a. Owing to this boundary condition, it is necessary to start with the unknown value p = a (recall that a = C,/KM (sensor)) rather than p = c (where c = C,/KM(bulk)). It should also be kept in mind that x = 0 at 1.1 = a and x = 1 when p = c. Thus we find from the definition of v above that l / v = 0 when p = a and eq 18 becomes 2a[a - log (1 a ) + p] = 0 (19) or p = -a log (1 + a) (20) Substituting for p in eq 18 l / v 2 = 2a[(p - a) - log (1 /A) + log (1 a)] (21)
-- = a [ p - log (1 + /I)
+
+
+
l/VZ
= 2a
[
(p - a) - log
'1
+ a + l
(22)
and
a l p l c .
This transformation crm be accomplished by the following sequence.
(7)
+
-u1= - =dP dx &[(p-a)-loga+l
+
']"'
(23)
so that
dx =
1 1
[
dp dp ( p - a) - log
p+112 a + l
(24)
1588
ANALYTICAL CHEMISTRY, VOL. 53, NO. 11, SEPTEMBER 1981
We can then integrate with x starting from x = 0 and p starting from p = a
(p -
At the point
p
= a we have
a) - log a + l
which gives
6a3a2 + 115'
a3a
($)a=---
(a
+ 115
(35)
(a
The series expansion (30) takes the form which is the exact solution for profile of x as a function of p. It should be recalled that x = X / L and p = C,/KM. At the boundary between the membrane and the bulk solution we have x = 1 and p = c, so that
P(X)
=
a + - - (YU x 2 a 1 2
+
+--+a2a
x4 (a + 1)s 24
- 6a3a2 x6
(a
+ 1)s
720
+
"'
(36) At the point x = 1 we have p(1) = c and the power series expansion becomes a2a a3a - 6a3a2 + + + ... 2(a + 1) 24(a +1)3 720(a + 1)5 CYU
c=a+-
which is the exact equation which determines the relationship between a and c, where a = C,/KM at the sensor and c = c,/KM in the bulk. The evaluation of the integral of eq 27 is not particularly easy but in the Appendix we show that it can be evaluated for either very small or very large values of the concentration a. By making use of the results of the Appendix, we find that for large a values we have (Y = 2b - 2[10g (1 C) - log (1 a)] (28) and for small a values we have
+
+
exp(&)
=
(p2 + 2p)1/2+ 0 + 1
p =b/a
The convergence of the series depends on the parameter
if s 1. In that case we may approximate the integral I(a;c)as I(a;c) =
[ + -][
I(a;c) = 2
a+c ac
: :I
fi[ 2 + - + (c - a ) - log
(~-1''~ l + c
(AB)
Anal. Chem. 1981, 53, 1590-1594
1590
We now consider the situation where a is very small. We substitute p - a = (a 1)s into eq A1 and we obtain
+
I(a;c) = (a
+ l ) S [ u s + s - log (1+ ~ ) ] - l / ~ d(A9) s
We plan to expand the integrand as a power series in a and assume that a
