A Mathematical Model for Predicting Dynamic ...

Copyright © 2015 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 12, 1–7, 2015

A Mathematical Model for Predicting Dynamic Sensitivity of a Non-Linear Amperometric Biosensor Model Noel Nesakumar1 2 , Uma Maheswari Krishnan1 3 , Swaminathan Sethuraman1 3 , and John Bosco Balaguru Rayappan1 2 ∗ 1

Centre for Nanotechnology and Advanced Biomaterials (CeNTAB), SASTRA University, Thanjavur 613401, Tamil Nadu, India 2 School of Electrical and Electronics Engineering, SASTRA University, Thanjavur 613401, Tamil Nadu, India 3 School of Chemical and Biotechnology, SASTRA University, Thanjavur 613401, Tamil Nadu, India Until now, various chemometric tools have been used to approximate solutions to address the nonlinear amperometric biosensor modeling challenge. Hence, an analytical method capable of predicting the non-constant slope (dynamic sensitivity) of a non-linear amperometric biosensor model has been developed. The dynamic sensitivity, and any given point in a given amperometric biosensor’s operation, is predicted by solving a series of mathematical constants corresponding to the physical entities of the amperometric biosensor itself. In order to validate the proposed models, data of butyrin biosensor were considered. The effectiveness of these models was statistically analyzed and found that the first-order dynamic sensitivity model was better than that of the second order. This dynamic sensitivity slope prediction, in turn, can be used to predict the concentration of an analyte when a non-constant amperometric sensor response is known.

Keywords: Amperometry, Dynamic Sensitivity, Percentage Recovery, Relative Prediction Error, Root Mean Square Error.

Accurate estimation of the response characteristics of a biosensor namely sensitivity, specificity, linearity, stability, biocompatibility, response time and dynamic response1 play a vital role in the quantification of biomarkers. Among all the response characteristics, sensitivity and dynamic response are highly significant in designing labon-a-chip and flow systems.2 3 For continuous monitoring of biomolecules in blood, biosensors should be calibrated at different times of a day.4 Laboratory scale calibrations are completely based on the linear response of the biosensor for known concentrations of biomolecules.5 Only an ideal biosensor can show such a linear response because of its dependence on the rate of diffusion and surface reaction.6 During calibration, most of the biosensors exposed to a known concentration of analyte show a dynamic response.3 But, in practice, the calibrated biosensors based on the linear response give inaccurate results because the concentrations of biomolecules fluctuate a great deal in between the calibrated values. Eva et al.3 reported that “handling non-linearity in the dynamic amperometric biosensor response is one of the future challenges.” They also specified that the dynamic response of ∗

Author to whom correspondence should be addressed.

J. Comput. Theor. Nanosci. 2015, Vol. 12, No. 6

an amperometric biosensor can be improved only when nonlinear interactions are added into the model. Until now, drift between the calibrated responses after exposing to the real time samples have been corrected using various chemometric tools7–10 by assuming that the change in the amperometric response between the real and reference samples are linear. But, there is a risk of change in sensitivity with time.3 And also, Meena et al.11 proposed an approximate analytical method by relating nonlinear time dependent amperometric system with the substrate via Michaelis-Menten kinetics. Even though the results seem promising, the complicated form of relationship between concentration of substrate, product and current makes the analytical method difficult to apply. In order to address the nonlinear amperometric biosensor modeling challenge and to reduce the complexity of Meena et al. theory, in this article, a simpler expression relating current, substrate and incubation time has been framed by replacing the complex attracting part. Until today, most of the nanointerface12–29 based biosensors reported sensitivity from the linear part of the nonlinear hyperbolic curve.30 This implied that sensitivity results obtained from the line-linear equation obeys the zero-order amperometric system. This may result in the improper estimation of sensitivity because Michaelis-Menten equation

1546-1955/2015/12/001/007

doi:10.1166/jctn.2015.3853

1

RESEARCH ARTICLE

1. INTRODUCTION

RESEARCH ARTICLE

A Mathematical Model for Predicting Dynamic Sensitivity of a Non-Linear Amperometric Biosensor Model

Nesakumar et al.

for an enzymatic reaction always follows a nonlinear hyperbolic curve. Alice et al.31 stated that the current response can be included in the linear range only when the standard residual of the current is lesser than two. Additionally, Rothwell et al.32 reported that an enzyme could access the substrate in linear fashion only when the substrate concentration is below one half of the Michaelis-Menten constant. On the other hand, Guilbault33 reported that the enzyme reaction could access the substrate in linear fashion only if its corresponding substrate concentration is under one fifth of the Michaelis-Menten constant and his prediction seems to be acceptable when compared with the existing methods for estimating linear range and sensitivity. This is valid in the case of a nanointerface based biosensor where the sensitivity remains constant only at the initial rate of enzymecatalyzed reaction. Practically, biosensors with an extended linear range need high enzyme immobilization.33 Further, the sensitivity measured from a linear curve reveals that the nanointerface based biosensor can respond quickly to a small change in substrate concentration.34 However, practically, it is difficult for a nanointerface based biosensor to respond instantaneously to a small change in substrate concentration because an enzyme takes minimum time to react with the substrate to attain a steady-state value.35 Moreover, incubation time, viscosity, specific activity of an enzyme and diffusion of substrate make an amperometric biosensor a time-dependent system. Hence it has become imperative to develop a mathematical model to estimate the exact dynamic response characteristics of a practical biosensor by considering both linear and nonlinear responses. With this background, first-order and second-order linear differential equations based mathematical models have been proposed to estimate the dynamic sensitivity (I /S, a changing nonconstant slope) of an amperometric biosensor.

where

2. MODEL DERIVATION

where Imax is the maximum current and KM is the Michaelis-Menten constant. Since the concentration of substrate is a function of incubation time, current of an amperometric biosensor can be written as

Generally, the dynamic response of a typical sensor can be represented by a differential equation. an

d n yt d n−1 yt + an−1 n dt dt n−1 dyt + a0 yt = F t + · · · + a1 dt

(1)

d m xt d m−1 xt dxt +b0 xt +b +· · ·+b1 m−1 m m−1 dt dt dt yt is the sensor output, xt is the sensor input, t is the incubation time and a and b are system constants. Based on this equation, the dynamic response of an amperometric biosensor can be represented in nth order ordinary differential equation, F t = bm

2

d n I t d n−1 I t + an−1 n dt dt n−1 dI t + a0 I t = F t + · · · + a1 dt

d m St d m−1 St dSt +b0 St +bm−1 +···+b1 m dt dt m−1 dt

I t is the current measured from amperometry, St is the input substrate concentration and a and b are system constants. In order to estimate the dynamic sensitivity of the biosensor, the order of the response has to be considered. Hence, the dynamic response of an amperometric system can be modeled using first-order linear differential equation. a1

dI t dSt + a0 I t = b1 + b0 St dt dt

(2)

(3)

Generally, in an amperometric system, sufficient incubation time is given to the substrate to get converted into the product. Assuming that the concentration of the substrate added at regular time intervals as a constant,36 37 one can write the relation between substrate concentration and incubation time as St = mt (4) where m is the constant (unit: Ms−1 Since the amperometric system is specific to output, the function b1 dSt/dt can be neglected a1

dI t + a0 I t = b0 mt dt

(5)

In an amperometric biosensor, the rate of an enzymecatalyzed reaction is measured in terms of current.37 I=

I t =

where

an

F t = bm

Imax S KM + S

Imax St KM + St

(6)

(7)

Considering the current as a function of incubation time and substituting Eqs. (4) in (7), we get, I t =

Imax mt KM + mt

(8)

Differentiating the current with respect to the incubation time, we get I mK M dI t = max (9) dt KM + mt2 Substituting Eqs. (8) and (9) in (5), we get Imax mKM Imax mt a1 = b0 mt + a 0 KM + mt KM + mt2

(10)

J. Comput. Theor. Nanosci. 12, 1–7, 2015

Nesakumar et al.


Generally, for a first-order system, the dynamic response can be modeled by a1

dyt + a0 yt = b0 xt dt

(11)

where a1 , a0 and b0 are constants, yt is the output from the system and xt is the input to the system. Solving the Eq. (11), the first-order dynamic sensitivity of a system can be found as K y = x D + 1

(12)

where K = b0 /a0 is the static sensitivity and = a1 /a0 is the systems time constant and D = d/dt Differentiating Eq. (7) with respect to the incubation time, the value of D can be calculated. KM D= tKM + mt

(13)

Substituting Eqs. (13) and (10) in (12) and solving, the first-order dynamic sensitivity of an amperometric biosensor can be deduced as I t b0 /a0 = St a1 /a0 KM /tKM + mt + 1 I t =

The dynamic response of an amperometric system can be modeled by second-order linear differential equation as follows: d 2 I t dI t a2 + a0 I t + a1 2 dt dt d 2 St dSt + b0 St = b2 2 + b1 d t dt

d 2 I t dI t + a0 I t = b0 St + a1 dt 2 dt

d 2 yt dyt + a1 + a0 yt = b0 xt 2 dt dt

where a0 , a1 , a2 and b0 are constant, yt is the output from the system and xt is the input to the system. Solving the Eq. (20), the second-order dynamic sensitivity of a system can be obtained as b0 /a0 y = 2 x a2 /a0 D + a1 /a0 D + 1

D2 =


−2KM m

(22)

tKM + mt2

Substituting Eqs. (22) and (13) in Eq. (21) and solving, the second-order dynamic sensitivity of an amperometric biosensor can be deduced as I t = b0 /a0 · a2 /a0 −2mK M /tKM + mt2 St + a1 /a0 KM /tKM + mt + 1 or + a1 /a0 KM /tKM + mt−1

(15)

(16)

+ 1 × St−1

= 100 × (17)

(18)

(23)

2.1. Statistical Analysis The performance of the proposed models was evaluated using percentage RSD, percentage recovery, root mean square error of cross validation, relative prediction error, total prediction error and regression coefficient (R2 . %recovery

Differentiating Eq. (9) once again with respect to substrate concentration, we get d 2 I t −2m2 K M Imax = dt 2 KM + mt3

(21)

where b0 /a0 is the static sensitivity, a1 /a0 and a2 /a0 are systems time constant, D = d/dt and D2 = d 2 /dt 2 . Differentiating Eq. (7) with respect to the t twice, the value of D2 can be calculated

Substituting Eqs. (4) in (16), we get d 2 I t dI t + a0 I t = b0 mt + a1 a2 2 dt dt

(20)

I t = b0 /a0 · a2 /a0 −2mK M /tKM + mt2

Since the amperometric system is specific to output, the functions b2 d 2 St/d 2 t and b1 dSt/dt can be neglected. a2

a2

n Dynamic sensitivity 1 ipredicted n i=1 Dynamic sensitivityiobserved

Root mean square error of cross validation n 1 = Dynamic sensitivityipredicted n i=1 05 2 − Dynamic sensitivityiobserved

(24)

(25) 3

RESEARCH ARTICLE

(14)

(19)

Generally for a second-order system, the dynamic response can be modeled by

or

b0 /a0 a1 /a0 KM /tKM + mt + 1 × st

Substituting Eqs. (8), (9) and (18) in (16), we get −2m2 K M Imax Imax mKM + a1 a2 KM + mt3 KM + mt2 Imax mt + a0 = b0 mt KM + mt


Relative prediction error n = Dynamic sensitivityipredicted

− Dynamic sensitivityiobserved ·

n

05 −1 2

Dynamic sensitivityipredicted

(26)

i=1

i=1

0.2

− Dynamic sensitivityiobserved

×

−1

0.8

1.0

1.2

1.4

1.6

1.8

Dynamic sensitivityipredicted

i=1 n

Dynamic sensitivityjpredicted

− Dynamic sensitivityjobserved ·

n

05 −1 2

Dynamic sensitivityjpredicted

(27)

j=1

RESEARCH ARTICLE

0.6

Fig. 1. Imax and KM of butyrin biosensor from Levenberg-Marquardt plot.

2

where Dynamic sensitivityiobserved is the ith observed dynamic sensitivity, Dynamic sensitivityipredicted is the ith predicted dynamic sensitivity estimated by prediction model, Dynamic sensitivityjpredicted is the jth predicted dynamic sensitivity estimated by prediction model and n is the number of samples. Matlab 6.5 software was used for the quantitative data analysis, since it is capable of nonlinear least squares fitting and can calculate errors of fitted enzyme kinetic parameters.

3. RESULTS AND DISCUSSION 3.1. Static and Dynamic Sensitivity Estimation The experimental data observed from the butyrin detecting amperometric biosensor (Table I) has been considered for the estimation of static and dynamic sensitivity.37 Imax and KM estimated from the Levenberg-Marquardt plot (Fig. 1) were found to be 119.937 A and 0.37 mM respectively. Figure 2 shows the static sensitivity measured (a) from linear regression (b) from the limit of linearity (c) up to Table I. Input data of butyrin detecting biosensor. Butyrin (mM)

4

0.4

Butyrin (mM)

2

j=1

0.33 0.66 0.99 1.32 1.65

Imax = 119.937 nA KM = 0.37 mM

60

2

·

80

70

Total prediction error n = Dynamic sensitivityipredicted

n

Incubation time - 20 min

90

2

Current (µΑ µΑ)

i=1

100

Nesakumar et al.

Current (nA) 58952 73542 85922 95428 98523

KM /2 and (d) up to KM /5. Static sensitivity estimated from linear regression and the limit of linearity remained the same, while the static sensitivity estimated from Imax /KM and up to KM /2 and KM /5 deviated a great deal. From this observation, one can understand the higher sensitivity and narrow linear range behavior of the developed butyrin biosensor only at the initial rate of lipase-butyrin interaction. The estimation of dynamic sensitivity after the initial rate of reaction will provide the information about extended dynamic range. Generally, within the linear range, the sensitivity of a biosensor for a small change in substrate concentration will remain a constant. This factor can be verified by estimating the R2 between the calculated and observed I. i.e., R2 = 1. From Table II, it was found that, when the butyrin concentration is below KM /5, the measured R2 between the observed and calculated I was approximately equal to one (≈ 099) while R2 approached nearly 0.98 for the butyrin concentration lesser than KM /2. These results showed that only up to KM /5, static sensitivity was observed. And, beyond KM /5, biosensor would show dynamic sensitivity towards the butyrin concentration. In order to prove the dynamic sensitivity of butyrin detecting amperometric biosensor, fractional saturation (Fig. 3(a)) was considered. Generally, fractional saturation represents the fraction of S that are bound to enzyme. It was found that at 0.33–1.65 mM concentrations of butyrin, nearly 47.14–81.68% of lipase’s active sites got occupied, which were greater than 16.66% fractional saturation obtained at 0.074 mM concentration of butyrin (onefifth of KM . And also at 0.33–1.65 mM concentrations of butyrin, the sensitivity of amperometric biosensor got decreased and showed dynamic behavior. This dynamic sensitivity of biosensor was due to the increase in binding of butyrin on to the active sites of lipase. Further, most of the actives sites of lipase occupied by butyrin decrease the J. Comput. Theor. Nanosci. 12, 1–7, 2015

Nesakumar et al.


(a) 100

1.5

90 85

Current (µΑ Α)

2.0 Sensitivity from limit of linearity = 36.91 nA mM–1

Regular residual output

95

(b) Sensitivity from –1 linear regression = 36.91 nA mM

80 75 70 65

1.0 0.5 0.0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

Butyrin (mM)

–0.5 –1.0

60 55

–1.5 0.4

0.6

0.8

1.0

1.2

1.4

Butyrin (mM) (c) 42

(d) –1

Sensitivity upto 0.5 KM = 195.755 nA mM

20

Sensitivity upto 0.2 KM = 267.207 nA mM–1

Current (µΑ Α)

Current (µΑ Α)

36

30

24

18

15

10

5

12 0.08

0.12

0.16

0.20

Butyrin (mM) Fig. 2.

0 0.00

0.02

0.04

0.06

0.08

Butyrin (mM)

Static sensitivity calculated (a) from linear regression (b) from the limit of linearity (c) up to KM /2 and (d) up to KM /5.

rate of lipase-catalyzed reaction and thus dynamic sensitivity was observed. 3.2. Model Validation Data of butyrin detecting amperometric biosensor was considered for the model validation in which the physical constant m is estimated to be 0.0165 mM min−1 (Fig. 3(b)). Using Eqs. (10) and (19), the unknown physical parameters a0 , a1 , a2 , b0 and b2 were calculated. Using, Eqs. (14) and (23), the first and second order dynamic sensitivities of butyrin detecting amperometric biosensor Table II.

Estimation of static sensitivity and its linear range.

Imax /KM Up to KM /2 Up to KM /5 From linear regression From limit of linearity

Linear range (mM)

Static sensitivity (nA mM−1 )

Regression coefficient

– Up to 0.185 Up to 0.074 0.33–1.32

324154 195755 267207 3691

– 0996 0988 0987

0.33–1.32

3691

0987


were computed and the results obtained are shown in Figure 3(c). Estimated dynamic sensitivity was compared with the predicted linear response because most of the biosensor research articles reported linearity using linear regression. The precision of both first and secondorder dynamic sensitivities was statistically analyzed using %RSD. To verify the reproducibility of the butyrin biosensor response, %RSD between observed and predicted amperometric response were calculated (Tables III and IV). The reproducibility of the butyrin biosensor response estimated using the first-order dynamic sensitivity model was found to be superior to the second-order dynamic (mean %RSD = 401%) and static sensitivity (1.17%) model as manifested by a mean %RSD of 1.045%. It showed that the butyrin biosensor response obeyed first-order dynamic response. It also evidenced that the first-order dynamic response can be used for butyrin detection over an extended linear range (0.33–1.65 mM) with good precision. From this observation, one can identify whether a biosensor response is following either first-order or second-order dynamic amperometric response. Table V shows the comparison of %recovery, root mean square 5

RESEARCH ARTICLE

0.04


Table III. Percentage relative standard deviation for predicted dynamic sensitivity values.

(a) 82.5

Fractional saturation (%)

Nesakumar et al.

75.0 Butyrin (mM)

67.5

0.33–0.66 0.66–0.99 0.99–1.32 1.32–1.65

60.0

Observed First-order Second-order %RSDa of %RSDa of dynamic dynamic dynamic first-order Second-order sensitivity sensitivity sensitivity dynamic dynamic −1 −1 −1 (nA mM ) (nA mM ) (nA mM ) sensitivity sensitivity 1459 1238 950 309

14803 12558 9645 3144

15684 12833 10212 3245

103 101 103 111

511 254 506 334

Note: a %RSD: Percentage relative standard deviation for predicted dynamic sensitivity values.

52.5

Table IV. Percentage relative standard deviation for predicted linear response values.

45.0 0.4

0.8

1.2

1.6

Butyrin (mM)

Butyrin (mM)

(b)

Observed amperometric response (nA)

Predicted response using linear regression (nA)

%RSDa

58952 73542 85922 95428

60189 72370 84551 96732

146 113 113 096

–1

m = 0.0165 mM min

Butyrin (mM)

1.6

0.33 0.66 0.99 1.32

1.2

Note: a %RSD: Percentage relative standard deviation for predicted linear response values.

0.8

20

40

60

80

100

Incubation time (min) (c) 15

∆I/∆S (nA mM– 1)

RESEARCH ARTICLE

0.4

12

9

6

Table V. Comparison of %recovery, root mean square error of cross validation, relative prediction error, total prediction error, regression coefficient and slope for different amperometric models.

Observed dynamic sensitivity First order dynamic sensitivity Second order dynamic sensitivity

First-order dynamic sensitivity

3 0.33 - 0.66

0.66 - 0.99

0.99 - 1.32

1.32 - 1.65

Butyrin (mM) Fig. 3. (a) Fractional saturation of lipase; (b) Estimation of physical parameter ‘m’ from linear regression and (c) Observed and simulated dynamic sensitivity of butyrin detecting biosensor.

error of cross validation, relative prediction error and total prediction error for the determination of the accuracy of different predicted butyrin amperometric response. The first-order dynamic response gave an accurate dynamic 6

sensitivity with root mean square error of cross validation of 0.1587, relative prediction error of 0.0145 and total prediction error of 8799 × 10−4 while the second-order dynamic sensitivity and static sensitivity results showed somewhat deviated figures of merit. The ability of the proposed models to recover the original butyrin biosensor response was validated using %recovery. Both the first and second order dynamic sensitivity values showed good %recovery in the range of 100.491–105.861%, and it was quite close to the %recovery (100.069) of the predicted linear response. These results confirmed the efficiency of the proposed dynamic sensitivity models. In addition to these results, R2 > 099 with a slope nearly equal to 1 (Table V) indicated that all predicted dynamic

Second-order Predicted dynamic response using sensitivity linear regression

Root mean square 0.1587 0.6951 error of cross validation Relative prediction 0.0145 0.0606 error %recovery 101.491 105.861 Total prediction 8799 × 10−4 error Regression 1.0 0.997 coefficient Slope ± Standard 1014 ± 259 × 10−4 1066 ± 0314 error

1.2727

0.0159 100.069

0.987 0991 ± 0065


Nesakumar et al.


amperometric sensitivity lies precisely on the observed dynamic amperometric sensitivity with minimal scatter. All the predicted dynamic sensitivities using the proposed models were seemed to indicate that the proposed models were quite acceptable and can measure the dynamic sensitivity of an amperometric biosensor for the determination of biomolecules in blood samples.

4. CONCLUSION

Instead, the extended linear range (0.33–1.65 mM) with slightly decreased sensitivity should be considered for better understanding the performance of a biosensor. Acknowledgments: The authors are grateful to the Nano Mission Council, (No. SR/NM/PG-16/2007).The authors also wish to express their sincere thanks to Department of Science and Technology, New Delhi for their financial support (DST/TSG/PT/2008/28), and (SR/FST/ETI-284/2011 (C)) for CH electrochemical analyzer. They also wish to acknowledge SASTRA University, Thanjavur for extending infrastructural support to carry out the study.

References 1. http://ocw.mit.edu/courses/materials-science-and-engineering/3-051jmaterials-for-biomedical-applications-spring-2006/lecture-notes/ lecture17.pdf. 2. N. Krishnaswamy, T. Srinivas, and G. M. Rao, IEEE Eng. Med. Biol. Soc. 2011, 30 (2011).

Received: 10 March 2014. Accepted: 11 April 2014.


7

RESEARCH ARTICLE

The two proposed models showed dynamic sensitivity similar to that of actual dynamic sensitivity of an amperometric biosensor. Investigation of the dynamic amperometric response model with the linear response and validation results based on %RSD, %recovery, relative prediction error, total prediction error and R2 showed that both the first and second order dynamic amperometric response models were effective and more accurate. Importantly, these models can estimate the concentration of biomolecules in the blood samples precisely, when the amperometric biosensors are exposed for continuous monitoring of biomolecules in blood samples. In addition, it is not advisable to judge the performance of a biosensor only based on the initial rate of an enzymatic reaction by calculating sensitivity from (i) KM /5, (ii) KM /2, (iii) linear regression or (iv) limit of linearity.

3. E. Dock, J. Christensen, M. Olsson, E. Tonning, T. Ruzgas, and J. Emneus, Talanta 65, 98 (2005). 4. N. K. Skjaervold, E. Solligard, D. R. Hjelme, and P. Aadahl, Anesthesiology 114, 120 (2011). 5. L. Luo, L. Zhu, and Z. Wang, Bioelectrochem. 88, 156 (2012). 6. P. B. Bansal, Bioanalysis and Biosensors in Agriculture Science, Oxford Book Company (2009), pp. 41–42. 7. J. E. Haugen, O. Tomic, and K. Kvaal, Anal. Chim. Acta 407, 23 (2000). 8. A. V. Lobanov, I. A. Borisov, S. H. Gordon, R. V. Greene, T. D. Leathers, and A. N. Reshetilov, Biosens. Bioelectron. 16, 1001 (2001). 9. G. A. Alonso, G. Istamboulie, T. Noguer, J. L. Marty, and R. Munoz, Sens. Actuators B 164, 22 (2012). 10. Y. Ni and S. Kokot, Anal. Chim. Acta. 626, 130 (2008). 11. A. Meena and L. Rajendran, J. Electroanal. Chem. 644, 50 (2010). 12. Srivastava, N. Jain, and A. K. Nagawat, Quantum Matter 2, 307 (2013). 13. G. H. Cocoletzi and N. Takeuchi, Quantum Matter 2, 382 (2013). 14. B. Tüzün and C. Erkoç, Quantum Matter 1, 136 (2012). 15. M. Narayanan and A. J. Peter, Quantum Matter 1, 53 (2012). 16. R. S. Sawhney, H. Kaur, and D. Engles, Quantum Matter 3, 127 (2014). 17. P. S. Yadav, D. K. Pandey, S. Agrawal, and B. K. Agrawal, Quantum Matter 3, 39 (2014). 18. Q. Zhao, Rev. Theor. Sci. 1, 83 (2013). 19. M. Ibrahim and H. Elhaes, Rev. Theor. Sci. 1, 368 (2013). 20. S. Celasun, Rev. Theor. Sci. 1, 319 (2013). 21. A. Herman, Rev. Theor. Sci. 1, 3 (2013). 22. P. D. Sia, Rev. Theor. Sci. 2, 146 (2014). 23. I. A. Sidorov, P. Prabakaran, and D. S. Dimitrov, J. Comput. Theor. Nanosci. 4, 1103 (2007). 24. V. A. Basiuk and M. Bassiouk, J. Comput. Theor. Nanosci. 5, 1205 (2008). 25. A. Groß, J. Comput. Theor. Nanosci. 5, 894 (2008). 26. C. Heitzinger, Y. Liu, N. J. Mauser, C. Ringhofer, and R. W. Dutton, J. Comput. Theor. Nanosci. 7, 2574 (2010). 27. J. A. T. Machado, D. Baleanu, E. Dinç, A. O. Solak, H. Eki, and R. Güzel, J. Comput. Theor. Nanosci. 8, 268 (2011). 28. Y. Haiying, Z. Jian, L. Jianjun, and Z. Junwu, J. Comput. Theor. Nanosci. 8, 1643 (2011). 29. E. Dinç and A. O. Solak, J. Comput. Theor. Nanosci. 11, 25 (2014). 30. H. Razmi, M. Harasi, Int. J. Electrochem. Sci. 3, 82 (2008). 31. A. C. Harper, M. R. Anderson, P. R. Carlier, G. L. Long, J. M. Tanko, and B. M. Tissue, Modified electrodes for amperometric determination of glucose and glutamate using mediated electron transport, Ph.D., Thesis, Blacksburg, VA (2005), pp. 1–152. 32. S. A. Rothwell, S. J. Killoran, and R. D. O’Neill, Sensors 10, 6439 (2010). 33. R. Baronas, F. Ivanauskas, and J. Kulys, Springer, New York (2010). 34. R. Devi, M. Thakur, and C. S. Pundir, Biosens. Bioelectron. 26, 3420 (2011). 35. Y. Liu, Z. Matharu, M. C. Howland, A. Revzin, and A. L. Simonian, Anal. Bioanal. Chem. 404, 1181 (2012). 36. K. Thandavan, S. Gandhi, S. Sethuraman, J. B. B. Rayappan, and U. M. Krishnan, Sens. Actuators B 176, 884 (2013). 37. S. Panky, K. Thandavan, D. Sivalingam, S. Sethuraman, U. M. Krishnan, and J. B. B. Rayappan, Mater. Chem. Phys. 137, 892 (2013).