ARTICLE pubs.acs.org/JPCA
Analytical Expression of Non-Steady-State Concentrations and Current Pertaining to Compounds Present in the Enzyme Membrane of Biosensor Anitha Shanmugarajan,† Subbiah Alwarappan,‡ and Lakshmanan Rajendran*,† † ‡
Department of Mathematics, The Madura College, Madurai 625011, India Nanotechnology Research and Education Center, University of South Florida, Tampa, Florida 33620-5350, United States ABSTRACT: A mathematical model of trienzyme biosensor at an internal diffusion limitation for a non-steady-state condition has been developed. The model is based on diffusion equations containing a linear term related to MichaelisMenten kinetics of the enzymatic reaction. Analytical expressions of concentrations and current of compounds in trienzyme membrane are derived. An excellent agreement with simulation data is noted. When time tends to infinity, the analytical expression of non-steady-state concentration and current approaches the steady-state value, thereby confirming the validity of the mathematical analysis. Furthermore, in this work we employ the complex inversion formula to solve the boundary value problem.
’ INTRODUCTION Biosensors are analytical devices that combine the selectivity and specificity of an immobilized biologically active compound with a signal transducer.13 The study of biosensors is of immense interest to optimize various analytical characteristics.48 The thickness of the enzyme membrane has a considerable effect on the biosensor response as well as on the response time.9,10 Practical biosensors that consist of a multilayer enzyme membrane,11 with the model biosensors containing the exploratory monolayer membrane, are useful tools to study the biochemical behavior of biosensors.12 Biosensors find immense applications in several fields, and this includes in routine clinical glucose monitoring,13 in agriculture and the food processing industry,14 and in the pharmaceutical industry.15 Earlier, analytical expressions pertaining to the steady-state concentrations and current of compounds in trienzyme membrane for all diffusion modules were calculated by Kulys.16 However, to the best of our knowledge, no analytical expressions pertaining to the non-steady-state concentrations and current of compounds for all diffusion modules R1d, R2d, and R3d in a trienzyme membrane have been reported. The purpose of this article is to arrive at analytical expressions corresponding to nonsteady-state concentrations and currents of compounds in trienzyme membrane for all diffusion modules using the complex inversion method. ’ MATHEMATICAL FORMULATION OF THE BOUNDARY VALUE PROBLEM In a creatinine biosensor, initially the hydrolyzed creatinine reacts with creatininase (E1) producing creatine (P1) and then r 2011 American Chemical Society
the hydrolyzed creatine reacts with creatinase (E2) producing sacrosine (P2). Following this, the oxidation of sacrosine will be carried out using sarcosine oxidase (E3) to yield hydrogen peroxide (P3). A general scheme that represents the reaction occurring at a creatinine biosensor is shown below: E1
E2
E3
S s f P1 s f P2 s f P3
ð1Þ
Here the rate of each reaction (Vi) is considered by the standard enzyme parameters Vmax(i) and Km(i), where i = 1, 2, and 3 for E1, E2, and E3 catalyzed processes, respectively. V1 = Vmax(1)s1/ Km(1), V2 = Vmax(2)p1/Km(2), and V3 = Vmax(3)p2/Km(3), where s1, p1, and p2 represent the concentrations of S, P1, and and P2 at concentrations less than the MichaelisMenten constants (Km(i)). The diffusion equations with a constant diffusion coefficient corresponding to the enzymatic conversion at a non-steady-state condition can be represented by eqs 25.16 1 Ds1 D2 s1 ¼ 2 R1 2 s1 D Dt Dx
ð2Þ
1 Dp1 D2 p1 ¼ 2 þ R1 2 s R2 2 p1 D Dt Dx
ð3Þ
1 Dp2 D2 p2 ¼ 2 þ R2 2 p1 R3 2 p2 D Dt Dx
ð4Þ
Received: January 18, 2011 Revised: March 29, 2011 Published: April 11, 2011 4299
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1 Dp3 D2 p3 ¼ 2 þ R3 2 p2 D Dt Dx
ð5Þ
D is the diffusion coefficient of all compounds; t is the time; Ri = (Vmax(i)/Km(i)D)1/2, where i = 1, 2, and 3; Km(i) denote MichaelisMenten constants; Vmax(i) denote the maximal enzymatic rates. The initial and boundary conditions are t ¼ 0, x ¼ d, x ¼ 0,
s1 ¼ s0 , s1 ¼ s0 , Ds1 ¼ 0, Dx
p1 ¼ 0,
p2 ¼ 0,
p1 ¼ 0,
p2 ¼ 0,
Dp1 ¼ 0, Dx
Dp2 ¼ 0, Dx
p3 ¼ 0
ð6aÞ
p3 ¼ 0
’ ANALYTICAL SOLUTIONS FOR THE CONCENTRATION AND CURRENT OF COMPOUNDS IN TRIENZYME MEMBRANE Using Laplace transformation in the partial differential equations eqs 25 and the condition eq 6a, the following differential equations in Laplace space are obtained:
¥ cosðlxÞ expððR 2 þ l2 ÞDtÞ 4s0 R1 4 1 πðR1 2 R2 2 Þ n ¼ 0 ð 1Þn ð2n þ 1ÞðR1 2 þ l2 Þ
4s0 R1 2 ¥ cosðlxÞ expððR2 2 þ l2 ÞDtÞ π n ¼ 0 ð 1Þn ð2n þ 1ÞðR2 2 R1 2 þ l2 Þ
∑
∑
¥ ð2n þ 1ÞcosðlxÞ expððR 2 þ l2 ÞDtÞ s 0 R1 4 π 2 d2 ðR1 2 R2 2 Þ n ¼ 0 ð 1Þn ðR2 2 R1 2 þ l2 Þ
∑
p2 ¼
s0 R1 2 R2 2 coshðR3 xÞ ðR3 2 R1 2 ÞðR3 2 R2 2 Þ coshðR3 dÞ
þ
¥ cosðlxÞ expððR 2 þ l2 ÞDtÞ 4s0 R1 4 R2 2 1 2 2 2 ðR1 R2 ÞðR1 R3 Þπ n ¼ 0 ð 1Þn ð2n þ 1ÞðR1 2 þ l2 Þ
þ
s0 R1 2 R2 2 coshðR2 xÞ ðR2 R1 2 ÞðR2 2 R3 2 Þ coshðR2 dÞ
þ
¥ ð2n þ 1Þ cosðlxÞ expððR 2 þ l2 ÞDtÞ s0 R1 4 R2 2 π 3 2 2 2 2 ðR1 R3 ÞðR2 R3 Þd n ¼ 0 ð 1Þn ðR3 2 þ l2 ÞðR3 2 R1 2 þ l2 Þ
þ
s0 R1 2 R2 2 coshðR1 xÞ ðR1 R2 2 ÞðR1 2 R3 2 Þ coshðR1 dÞ
¥ ð2n þ 1Þ cosðlxÞ expððR 2 þ l2 ÞDtÞ s0 R1 4 R2 2 π 2 ðR1 2 R2 2 ÞðR2 2 R3 2 Þd2 n ¼ 0 ð 1Þn ðR2 2 þ l2 ÞðR2 2 R1 2 þ l2 Þ
þ
4s0 R1 2 R2 2 ¥ ð 1Þn cosðlxÞ expððR3 2 þ l2 ÞDtÞ 2 2 2 2 2 2 π n ¼ 0 ð2n þ 1ÞðR3 R1 þ l ÞðR3 R2 þ l Þ
∑
2
2
∑
2
2
∑
d2 s1 s s0 R1 2 s 1 s 1 þ ¼ 0 2 D dx D
ð8Þ
d 2 p1 s þ R1 2 s 1 R2 2 p1 p1 ¼ 0 2 D dx
ð9Þ
þ
4s0 R1 2 R2 2 ¥ cosðlxÞ expððR2 2 þ l2 ÞDtÞ ðR2 2 R3 2 Þπ n ¼ 0 ð 1Þn ð2n þ 1ÞðR2 2 R1 2 þ l2 Þ
d2 p2 s þ R2 2 p1 R3 2 p2 p2 ¼ 0 D dx2
ð10Þ
s0 R1 2 R2 2 π ¥ ð2n þ 1Þ cosðlxÞ expððR3 2 þ l2 ÞDtÞ ðR2 2 R3 2 Þd2 n ¼ 0 ð 1Þn ðR3 2 R1 2 þ l2 ÞðR3 2 R2 2 þ l2 Þ
d2 p3 s þ R3 2 p2 p3 ¼ 0 2 D dx
ð11Þ
p3 ¼
Now the boundary conditions become s0 x ¼ d, s1 ¼ ; p1 ¼ 0; p2 ¼ 0; s ds1 ¼ 0; dx
dp1 ¼ 0; dx
dp2 ¼ 0; dx
p3 ¼ 0
ð12aÞ
s0 coshðR1 xÞ 4s0 R1 þ coshðR1 dÞ π
2 ¥
cosðlxÞ expððR1 þ l ÞDtÞ n 2 2 n ¼ 0 ð 1Þ ð2n þ 1ÞðR1 þ l Þ
∑
2
2
ð13Þ
∑
∑
∑
ð15Þ
s0 R1 2 R2 2 ð1 coshðR3 xÞÞ ðR3 R1 2 ÞðR3 2 R2 2 Þ coshðR3 dÞ 2
þ
¥ ð 1Þn sinðmðd xÞÞ expðm2 DtÞ 2s0 R1 4 R2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πðR1 2 R3 2 ÞðR2 2 R3 2 Þ n ¼ 1 nðR1 2 m2 Þ coshð R3 2 m2 dÞ
s0 R1 2 R3 2 ð1 coshðR2 xÞÞ ðR1 R2 2 ÞðR2 2 R3 2 Þ coshðR2 dÞ
¥ ð 1Þn sinðmðd xÞÞ expðm2 DtÞ 2s0 R1 4 R3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 πðR1 R2 ÞðR2 R3 Þ n ¼ 1 nðR1 2 m2 Þ coshð R2 2 m2 dÞ
þ
s0 R2 2 R3 2 ð1 coshðR1 xÞÞ ðR1 2 R2 2 ÞðR1 2 R3 2 Þ coshðR1 dÞ
þ
¥ ð 1Þn sinðmðd xÞÞ expðm2 DtÞ 2s0 R1 2 R2 2 R3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 πðR1 R2 ÞðR1 R3 Þ n ¼ 1 nðR1 2 m2 Þ coshð R1 2 m2 dÞ
þ
s 0 R1 2 R2 2 x 1 1 dðR3 2 R1 2 ÞðR3 2 R2 2 Þ coshðR3 dÞ
p3 ¼ 0 ð12bÞ
where s is the Laplace variable and an overbar indicates a Laplacetransformed quantity. The set of expressions presented in eqs 811 defines the initial and boundary value problem in Laplace space. The analytical solutions (see Appendix A) of eqs 25 are s1 ¼
ð14Þ
Dp3 ¼ 0 ð6cÞ Dx
where F is the Faraday number.
x ¼ 0,
þ
ð6bÞ
where d is the membrane thickness. The biosensor current i is defined by the following expression: Dp3 i ¼ 2FD ð7Þ Dx x ¼ 0
s0 R1 2 coshðR2 xÞ coshðR1 xÞ R1 2 R2 2 coshðR2 dÞ coshðR1 dÞ
p1 ¼
4300
∑
2
2
2
∑
∑
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þ
2s0 R1 2 R2 2 π ¥ nð 1Þn sinðmðd xÞÞ expðm2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðR2 R3 Þd n ¼ 1 ðR1 2 m2 ÞðR2 2 m2 Þ coshð R3 2 m2 dÞ
s 0 R 1 2 R3 2 x 1 1 dðR1 2 R2 2 ÞðR2 2 R3 2 Þ coshðR2 dÞ
∑
þ
þ
¥ 2s0 R1 4 R2 2 expð m2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 dðR1 R3 ÞðR2 R3 Þ n ¼ 1 ðR1 m2 Þ coshð R3 2 m2 dÞ
∑
2
" þ
2s0 R1 2 R3 2 π ¥ nð 1Þn sinðmðd xÞÞ expðm2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR2 2 R3 2 Þd2 n ¼ 1 ðR1 2 m2 ÞðR2 2 m2 Þ coshð R2 2 m2 dÞ
∑
þ
þ
2s0 R1 4 R2 2 2s0 R1 4 R3 2 2 2 2 2 dðR1 R3 ÞðR2 R3 Þ dðR1 R2 2 ÞðR2 2 R3 2 Þ 2
¥
n
ð 1Þ expð m 2s0 R1 2 R2 2 R3 2 þ dðR1 2 R2 2 ÞðR1 2 R3 2 Þ n ¼ 1 R1 2 m2
s 0 R2 2 R3 2 x 1 1 2 2 2 dðR1 R2 ÞðR1 R3 Þ coshðR1 dÞ 2
þ
2s0 R1 2 R2 2 R3 2 ¥ ðsinðmxÞ þ sinðmðd xÞÞÞ expðm2 DtÞ n 2 2 2 2 2 2 π n ¼ 1 nð 1Þ ðR1 m ÞðR2 m ÞðR3 m Þ
∑
∑
2
DtÞ
2s0 R1 2 π2 ¥ n2 ð 1Þn expðm2 DtÞ 2 2 2 2 d3 n ¼ 1 ðR1 m ÞðR2 m Þ
∑
2s0 R1 2 R2 2 π2 ¥ n2 expðm2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m2 ÞðR 2 m2 ÞðR 2 m2 Þ coshð R 2 m2 dÞ d3 ðR 1 2 3 3 n¼1
∑
2s0 R1 2 R2 2 2s0 R1 2 R3 2 2 2 2 2 πðR3 R1 ÞðR3 R2 Þ πðR1 R2 2 ÞðR2 2 R3 2 Þ 2
2s0 R2 2 R3 2 πðR1 R2 2 ÞðR1 2 R3 2 Þ
2
¥
∑ n¼1
ð17Þ
ðsinðmxÞ expðm2 DtÞ nð 1Þn ðR1 2 m2 Þ
where l¼
2s0 R1 2 π ¥ n sinðmxÞ expðm2 DtÞ þ n 2 2 2 2 2 d n ¼ 1 ð 1Þ ðR1 m ÞðR2 m Þ
∑
þ
2s0 R1 2 R2 2 π ¥ ð 1Þn n sinðmxÞ expðm2 DtÞ 2 2 2 2 2 2 d2 n ¼ 1 ðR1 m ÞðR2 m ÞðR3 m Þ
þ
2s0 R1 2 R2 2 π ¥ nð 1Þn sinðmðd xÞÞ expðm2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 2 2 d n ¼ 1 ðR1 m ÞðR2 m ÞðR2 m Þ coshð R3 m dÞ
∑
∑
ð16Þ When time tends to infinity, the analytical expression of nonsteady-state concentration (eqs 1316) approaches the steadystate value, thereby confirming the validity of the mathematical analysis. The biosensor current i is (
2s0 R1 2 R2 2 π2 ¥ n2 ð 1Þn expðm2 DtÞ i ¼ 2FD 2 m2 ÞðR 2 m2 ÞðR 2 m2 Þ d3 ðR 1 2 3 n¼1
∑
¥
2s0 R1 R2 R3 expðm DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðR1 2 R2 2 ÞðR1 2 R3 2 Þ n ¼ 1 ðR1 2 m2 Þ coshð R1 2 m2 dÞ
þ
2s0 R1 2 R2 2 R3 2 ¥ ðð 1Þn 1Þ expð m2 DtÞ 2 2 2 2 2 2 d n ¼ 1 ðR1 m ÞðR2 m ÞðR3 m Þ
þ
2s0 R1 2 R3 2 π2 ¥ n2 expð m2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d3 ðR2 2 R3 2 Þ n ¼ 1 ðR1 2 m2 ÞðR2 2 m2 Þ coshð R1 2 m2 dÞ
þ
s0 R1 2 R2 2 1 1 dðR3 2 R1 2 ÞðR3 2 R2 2 Þ coshðR3 dÞ
2s0 R1 2 R2 2 π2 ¥ n2 expð m2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 2 2 d ðR2 R3 Þ n ¼ 1 ðR1 m ÞðR2 2 m2 Þ coshð R1 2 m2 dÞ
s0 R1 2 R3 2 1 1 dðR1 2 R2 2 ÞðR2 2 R3 2 Þ coshðR2 dÞ
þ
¥ 2s0 R1 4 R3 2 expð m2 DtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðR1 2 R2 2 ÞðR2 2 R3 2 Þ n ¼ 1 ðR1 2 m2 Þ coshð R2 2 m2 dÞ
þ
s0 R2 2 R3 2 1 1 dðR1 2 R2 2 ÞðR1 2 R3 2 Þ coshðR1 dÞ
2
2
2
∑
2
∑
∑
∑
∑
ð2n þ 1Þπ d
and
m¼
nπ d
ð18Þ
When t f ¥, the steady-state current was obtained from eq 17 and is shown by ( s0 R1 2 R2 2 1 1 i ¼ 2FD coshðR3 dÞ dðR3 2 R1 2 ÞðR3 2 R2 2 Þ s 0 R1 2 R3 2 1 1 coshðR2 dÞ dðR2 2 R1 2 ÞðR3 2 R2 2 Þ 2 2 s 0 R2 R3 1 1 þ ð19Þ coshðR1 dÞ dðR21 R2 2 ÞðR1 2 R3 2 Þ This steady-state current is same as the result of the work of Kulys [see eq 7 of ref 16].
’ NUMERICAL SIMULATION The diffusion equations [eqs 25] for the boundary conditions [eqs 6a6c] are solved by numerical methods. The function pdex4 in Matlab software, which is a function of solving the initial boundary value problems for partial differential equations, was used to solve these equations. Its numerical solution is compared with our analytical results in Figure 1, and it gives a excellent agreement for all values of time t and diffusion modules R1d, R2d, and R3d in a trienzyme membrane. The Matlab program is also given in Appendix C. ’ DISCUSSION Figure 1 represents the non-steady-state concentration profiles in enzyme membrane of biosensor. Figure 1a shows the concentration s1 versus membrane thickness x in trienzyme membrane for various values of time t and for diffusion modules R1d, R2d, and R3d. From Figure 1a, it is evident that the value of the concentration s1 ≈ 1 when x = 0.01. The value of the concentration s1 decreases when time increases. The concentration reaches the steady-state value when t g 10. Figure 1b represents the concentration p1 versus membrane thickness x in trienzyme membrane for various values of time t and for diffusion modules R1d, R2d, and R3d. From Figure 1b, it is inferred that when x e 0.004, all the curves reach the 4301
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Figure 1. Concentration profiles for (a) concentration s1, (b) concentration p1, (c) concentration p2, and (d) concentration p3 in trienzyme membrane for various values of time t when s0 = 1 μmol/cm3, d = 0.01 cm, and for diffusion modules R1d = 10.0, R2d = 10.1, and R3d = 10.3. Symbols: (—) eqs 1316; (þ) numerical simulation.
Figure 2. Plot of compound concentration versus membrane thickness x in trienzyme membrane. The concentrations were computed using eqs 1316 for s0 = 1 μmol/cm3, R1d = 10.0, R2d = 10.1, R3d = 10.3, d = 0.01 cm, and time t g 10. Symbols: (—) eqs 1316; (þ) Figure 1 of ref 16.
Figure 3. Plot of the three-dimensional current i calculated using eqs 17 and 19 for s 0 = 1 μmol/cm3, R3d = 10.3, and t g 100. 4302
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Figure 4. Biosensor current i plotted as a function of time t. The biosensor currents were computed using eq 17 when s0 = 1 μmol/cm3 for (a) fixed values of R1d = 0.1 and R2d = 1 and various values of R3d, (b) fixed values of R1d = 1 and R3d = 0.5 and various values of R2d, and (c) fixed values of R2d = 1 and R3d = 1.01 and various values of R1d.
steady-state value. A series of non-steady-state concentrations p2 for various values of time t is plotted in Figure 1c. From Figure 1c, it is obvious that there is a simultaneous decrease in the values of concentration p2 and the steady-state value is reached when t g 10. The non-steady-state concentration p3 for various values of time t and for diffusion modules R1d, R2d, and R3d is plotted in Figure 1d. From Figure 1d, it is evident that initially the value of the concentration increases and reaches the maximum value when x = 0.006 and t = 5. The value of the concentration p3 becomes zero when x = 0 and x = 0.01. Figure 2 represents the non-steady-state concentrations s1, p1, p2, and p3 versus membrane thickness x in trienzyme membrane for various values of time t and for diffusion modules R1d, R2d, and R3d. In Figure 2, our non-steady-state analytical results [eqs 1316] are compared with steady-state results when t g 10, and further when x g 0.006 the value of concentration s1 increases and reaches the value 1. The value of the concentration p1 increases when x g 0.004 and decreases gradually to reach the value 0.01. The value of the concentration p2 increases when x g 0.004 and decreases gradually to reach the value 0.01. The value of the concentration p3 increases and attains the maximum value when 0.006 e x e 0.007 and then decreases gradually. Figure 3 represents a three-dimensional current i calculated using eqs 17 and 19 for s0 = 1 μmol/cm3, R3d = 10.3, and t g 100. Figure 3 compares the biosensor response i obtained in this work with Kulys's work.16 Upon comparison, it is evident that both results are
identical. Figure 4 indicates the non-steady-state current response i for various values of diffusion modules R1d, R2d, and R3d. Figure 4a represents the current i for all values of R3d [=1.01, 1.1, 1.5, 2] when R1d = 0.1 and R2d = 1 using eq 17. From Figure 4a, it is evident that i increases when R3d increases. Furthermore, when t g 20, all the curves reach the steady-state value. Figure 4b indicates the current i for all values of R2d [=1.01, 1.1, 1.5, 2] when R1d = 1 and R3d = 0.5 using eq 17. From Figure 4b, it is evident that i increases when R2d increases. Also, when t g 20, all the curves reach the steady-state value. Figure 4c indicates the current i for all values of R1d [=1.03, 1.1, 1.5, 1.8] when R2d = 1 and R3d = 1.01 using eq 17. From Figure 4c, it is evident that i increases when R3d increases. Also, when t g 20, all the curves reach the steady-state value.
’ CONCLUSIONS In this work, we obtained an analytical solution for a biosensor containing three immobilized enzymes utilizing consecutive substrate conversion. An amperometric trienzyme biosensor electrode was constructed for the determination of creatine and creatinine in pharmaceutical products and serum samples. The response characteristics of the electrodes are measured at different potentials to determine the appropriate working potential (higher sensitivity, lower limit of detection, wide linear concentration range, etc.) for the simultaneous detection of creatine and creatinine. Time-dependent linear diffusion equations have been formulated and solved analytically. The analytical expressions for the concentrations and 4303
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current for catalytic reactions with diffusion coefficient at trienzyme membrane under non-steady-state conditions are obtained by using the complex inversion formula. An excellent agreement with the existing steady-state result is noted. Furthermore, the numerical simulation was in excellent agreement with the analytically obtained concentration and current.
’ APPENDIX A Solution of eqs 811 Using Complex Inversion Formula. In this appendix we indicate how eqs 1316 are derived. By solving a differential equation of second order with constant coefficients, we obtained the solution of the eq 8 as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s s0 R1 2 D cosh R21 þ x D s0 s1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! þ s þ R 2 D 1 s sðs þ R1 2 DÞ cosh R1 2 þ d D
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s sðs þ R1 DÞ cosh R1 2 þ d ¼ 0 D 2
Hence there is a simple pole at s = 0 and there is a pole at 2 s = R1 D, and there are infinitely many poles given by the solution of the equation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s cosh R1 2 þ d ¼ 0 D and so
ðA1Þ sn ¼ R1 2
In this appendix we indicate how eq A1 may be inverted using the complex inversion formula. If y(s) represents the Laplace transform of a function y(τ), then according to the complex inversion formula we can state that Z 1 c þ i¥ exp½sτ yðsÞ ds yðτÞ ¼ 2πi c i¥ I 1 ¼ exp½sτ yðsÞ ds ðA2Þ 2πi c where the integration in eq A2 is to be performed along a line s = c in the complex plane where s = x þ iy. The real number c is chosen such that s = c lies to the right of all the singularities, but is otherwise assumed to be arbitrary. In practice, the integral is evaluated by considering the contour integral presented on the right-hand side of eq A2, which is evaluated using the so-called Bromwich contour. The contour integral is then evaluated using the residue theorem which states for any analytic function F(z) I FðzÞ dz ¼ 2πi Re s½FðzÞz ¼ zn ðA3Þ c
Now the poles are obtained from
3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s cosh R1 2 þ x 7 6 7 6 D 7 6 ! SðX, TÞ ¼ Re s6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 6 s 5 4 sðs þ R1 2 DÞ cosh R1 2 þ d D 2
3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s 2 cosh R1 þ x 7 6 7 6 D 7 6 ! þ Re s6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 6 s 5 4 sðs þ R1 2 DÞ cosh R1 2 þ d D
n
zs fa
Hence in order to invert eq A1, we need to evaluate 3 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s cosh R1 2 þ x 7 6 7 6 D 7 6 ! Re s6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 6 s 5 4 sðs þ R1 2 DÞ cosh R1 2 þ d D
ðA6Þ
s ¼ sn
The first residue in eq A6 is given by 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s 2 cosh R1 þ x 7 6 7 6 D 7 6 ! Re s6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 6 s 5 4 sðs þ R1 2 DÞ cosh R1 2 þ d D 2
s¼0
8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 9 > > s > > > R1 2 þ x > > > = < expðsTÞ cosh D ! ¼ lim rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ss f0 > > s > > 2 DÞ cosh > R1 2 þ d > ðs þ R > > 1 : D ;
ðA4Þ
From the theory of complex variables we can show that the residue of a function F(z) at a simple pole at z = a is given by Re s½FðzÞz ¼ a ¼ lim fðz aÞFðzÞg
s¼0
2
∑n
∑n Re s½exp½sτ yðsÞs ¼ s
with n ¼ 0, 1, 2, :::
Hence we note that
where the residues are computed at the poles of the function F(z). Hence from eq A3, we note that yðτÞ ¼
ð2n þ 1Þ2 π2 4d2
¼
ðA5Þ
coshðR1 xÞ R1 coshðR1 dÞ
ðA7Þ
2D
The second residue in eq A6 is as follows: 3 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s 2 cosh R1 þ x 7 6 7 6 D 7 6 ! Re s6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 7 6 s 5 4 sðs þ R1 2 DÞ cosh R1 2 þ d D
s¼0
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8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !9 > > s > > >expðsTÞ cosh R1 2 þ x > > = < D > ! ¼ lim 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > ss f R1 D > s > > 2þ > > s cosh R d > > 1 ; : D ¼
expð R1 2 DTÞ R1 2 D
t=1; s=0; N=100; for n=0: 1 :Nþ1; s=sþ(-1)∧n*cos((2*nþ1)*pi*x/(2*d)) *exp(-alpha1∧2*D*tþ(2*nþ1)^2*pi∧2*D*t/(4*d∧2)))/ ((2*nþ1)*(alpha1∧2þ(2*nþ1)∧2*pi∧2/(4*d∧2))); end u=m*cosh(alpha1*x)/cosh(alpha1*d); v=uþ4*m*alpha1∧2/pi*s; plot(x,v) Similarly, we can find the sum of the series of eqs 1416.
ðA8Þ
The third residue in eq A6 can be evaluated as follows. It is established that if F(z) can be expressed as F(z) = f(z)/g(z), where the functions f and g are analytic at s = sn and g(sn) = 0 while g0 (sn) 6¼ 0 and f(sn) 6¼ 0, then Re s½FðzÞs ¼ sn ¼
¥
f ðs Þ
∑ 0 n exp½sτ n ¼ 0 g ðsn Þ
Hence we can show that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s cosh R1 2 þ x D Lt est rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ss fsn d s sðs þ R1 2 DÞ cosh R1 2 þ d ds D rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# s s cosh R1 2 þ x 2D R1 2 þ D D ¼ Lt est rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ss fsn s sn dðs þ R1 2 DÞ cosh R1 2 þ d D "
! # ð2n þ 1Þ2 π2 ð2n þ 1Þπ ð2n þ 1Þπ cosh i x 2D exp R1 þ Dt i 2d 2d 4d2 ! ¼ 2 2 2 2 ð2n þ 1Þ π ð2n þ 1Þ π ð2n þ 1Þπ d d R1 2 þ D sinh i D 2 2 2d 4d 4d 2
for ¼ 0, 1, :::
ðA9Þ
Using cosh(iθ) =cos (θ) and sinh(iθ) = i sin(θ)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! s R1 2 þ x D Lt est rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ss fs n d s sðs þ R1 2 DÞ cosh R1 2 þ d ds D " ! # ð2n þ 1Þπ ð2n þ 1Þ2 π2 n 2 x exp R1 þ ð 1Þ cos Dt 2d 4d2 4 ¥ ! ¼ πD n ¼ 0 ð2n þ 1Þ2 π2 ð2n þ 1Þ R1 2 þ 4d2 cosh
∑
ðA10Þ Similarly, we can invert eqs 911.
’ APPENDIX B This appendix contains the Matlab program to find the sum of the series of eq 13. function v=v(x) alpha1=1000; m=1; d=0.01; D=3*10∧(-6); x=linspace(0,0.01);
’ APPENDIX C The Matlab program to find the numerical solution of eqs 25 is as follows. function pdex4 m=0; x=linspace(0,0.01); t=linspace(0,0.5); sol=pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u1=sol(:,:,1); u2=sol(:,:,2); u3=sol(:,:,3); u4=sol(:,:,4); figure plot(x,u1(end,:)) title( 0 u1(x,t) 0 ) figure plot(x,u2(end,:)) title( 0 u2(x,t) 0 ) figure plot(x,u3(end,:)) title( 0 u3(x,t) 0 ) figure plot(x,u4(end,:)) title( 0 u4(x,t) 0 ) % -------------------------------------------------------------function [c,f,s]=pdex4pde(x,t,u,DuDx) D=3*10∧(-6); alpha1=1000; alpha2=1010; alpha3=1030; c=[1/D; 1/D; 1/D; 1/D]; f=[1; 1; 1; 1].* DuDx; F1=-alpha1∧2*u(1); F2=alpha1∧2*u(1)-alpha2∧2*u(2); F3=alpha2∧2*u(2)-alpha3∧2*u(3); F4=alpha3∧2*u(3); s=[F1; F2; F3; F4]; % -------------------------------------------------------------function u0=pdex4ic(x); u0=[1; 0; 0; 0]; % -------------------------------------------------------------function [pl,ql,pr,qr]=pdex4bc(xl,ul,xr,ur,t) pl=[0; 0; 0; ul(4)]; ql=[1;1; 1; 0]; pr=[ur(1)-1; ur(2); ur(3); ur(4)]; qr=[0; 0; 0; 0]; 4305
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’ AUTHOR INFORMATION Corresponding Author
*Tel.: þ91 9442228951. Fax: þ91 0452 2675238. E-mail: raj_sms@rediffmail.com.
’ ACKNOWLEDGMENT This work was supported by the CSIR, New Delhi, India. The authors are thankful to Dr. T. V. Krishnamoorthy, The Prinicipal, The Madura College, Madurai, and Mr. M. S. Meenakshisundaram, The Secretary, Madura College Board, Madurai, for their encouragement. We thank the reviewers for their valuable comments to improve the quality of the manuscript. ’ REFERENCES (1) Guilbault, G. G. Analytical Uses of Immobilized Enzymes; Marcel Dekker: New York, 1984. (2) Biosensors: Fundamentals and Applications; Turner, A. P. F., Karube, I., Wilson, G. S., Eds.; Oxford University Press: Oxford, U.K., 1987. (3) Scheller, F.; Schubert, F. Biosensors; Elsevier: Amsterdam, 1992. (4) Guilbault, G. G.; Nagy, G. Anal. Chem. 1973, 45, 417–419. (5) Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 47, 299–307. (6) Mell, L. D.; Maloy, J. T. Anal. Chem. 1976, 48, 1597–1601. (7) Kulys, J. J.; Sorochinski, V. V.; Vidziunaite, R. A. Biosensors 1986, 2, 135–146. (8) Schulmeister, T.; Rose, J.; Scheller, F. Biosens. Bioelectron. 1997, 12, 1021–1030. (9) Schulmeister, T. Sel. Electrode Rev. 1990, 12, 203–260. (10) Baronas, R.; Ivanauskas, F.; Kulys, J.; Sapagovas, M. Nonlinear Anal.: Modell. Control 2003, 8, 3–18. (11) Baeumner, A. J.; Jones, C.; Wong, C. Y.; Price, A. Anal. Bioanal. Chem. 2004, 378, 1587–1593. (12) Baronas, R.; Ivanauska, F.; Kulys, J. Sensors 2003, 3, 248–262. (13) Velho, G. D.; Reach, G.; Thevenot, D. R. The design and development of in vivo glucose, sensors for an artificial endocrine pancreas. In Biosensors: Fundamentals and Applications; Turner, A. P. F., Karube, I., Wilson, G. S., Eds.; Oxford University Press: New York, 1987. (14) Whitaker, J. R. The need for biosensors in the food industry and food research. In Food Biosensor Analysis; Wagner, G., Guilbault, G. G., Eds.; Marcel Dekker: New York, 1994. (15) Stefan, R. I.; Bokretsion, R. G.; Staden, J. F.; Aboul-Enein, H. Y. Talanta 2003, 60, 1223–1228. (16) Kulys, J. Nonlinear Anal.: Modell. Control 2004, 9, 139–144.
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