#
Analysis of Stochastic Time Response of Microfluidic Biosensors in the Case of Competitive Adsorption of Two Analytes^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{1}, N

_{2}), where N

_{1}and N

_{2}are random processes, representing the number of adsorbed particles of substances 1 and 2, respectively, at the moment t. Assuming that in the time interval dt→0 the change of the number of adsorbed particles of only one substance is possible, and at most by 1, the transition rates between adjacent states are: A

_{1}(N

_{1}, N

_{2}), denoting the probability of increase of the number of type 1 adsorbed particles per unit time, D

_{1}(N

_{1}) denoting the probability of decrease of N

_{1}by 1 per unit time, and the corresponding probabilities for N

_{2}are A

_{2}(N

_{1}, N

_{2}) and D

_{2}(N

_{2}), respectively. Assuming adsorption of one particle to one binding site, uniformity of all binding sites, and the absence of interaction between analyte particles, the use of TCM for approximation of the spatial distribution of time dependent concentrations of both particle types in the reaction chamber yields the expressions

_{1}and C

_{2}are concentrations of two substances in the analyzed sample, k

_{a}

_{1}and k

_{a}

_{2}are their adsorption rate constants, k

_{d}

_{1}and k

_{d}

_{2}are desorption rate constants, k

_{m}

_{1}and k

_{m}

_{2}are mass transfer coefficients (they model particle transfer by both convection and diffusion between the bulk solution and binding sites, according to TCM), and N

_{m}is the total number of binding sites on the sensing surface.

_{1}> and <N

_{2}>, their variances, σ

_{1}

^{2}and σ

_{2}

^{2}, and the covariance, σ

_{12}, and by using the master equation for the probability of states of the bivariate random process, the system of five equations is obtained for the mentioned first and second moments (the nonlinear transition rates (Equations (1) and (2)) are approximated by the Taylor series; all derivatives are calculated for N

_{1}= <N

_{1}> and N

_{2}= <N

_{2}>)

(4) | |

(5) | |

(6) | |

(7) |

_{1}and m

_{2}are the weight factors, which represent the average contribution of a single particle of the first and second analyte to the sensor response).

## 3. Results and Discussion

_{1}= 1 nM, k

_{a}

_{1}= 8 × 10

^{7}1/(Ms), k

_{d}

_{1}= 0.08 1/s, k

_{m}

_{1}= 2 × 10

^{−5}m/s C

_{2}= 2 nM, k

_{a}

_{2}= 8 × 10

^{6}1/(Ms), k

_{d}

_{2}= 0.02 1/s, k

_{m}

_{2}= 2 × 10

^{−5}m/s, A = 1 × 10

^{−9}m

^{2}, and the adsorption sites surface density n

_{m}= N

_{m}/A = 1 × 10

^{−11}Mm (1 M = 1 mol/dm

^{3}). Figure 1b shows the expected number of adsorbed particles when only the target analyte exists in the analyzed sample with the same concentration C

_{1}(the stochastic response model for a single analyte is presented in [3]). The influence of competitive adsorption on the change of the number of adsorbed target analyte particles is pronounced, and it is quantitatively determined based on the shown diagrams.

_{D}

_{1}, N

_{D}

_{2}), which takes into account mass transfer [5], are denoted by dashed lines, for a sensing area A = 1 × 10

^{−14}m

^{2}with N

_{m}= 6000 binding sites, C

_{1}= 50 pM, C

_{2}= 100 pM, and the same values of other parameters as for Figure 1. A significant difference between the kinetics predicted by the deterministic model and by the more accurate stochastic model of sensor response can be seen. It is interesting to note that the two models give an opposite prediction regarding the analyte that is dominantly adsorbed.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Time dependent expected value of the adsorbed particles number of the target (red line) and the competitor analyte (blue line), obtained by using the stochastic response model; (

**b**) Expected number of adsorbed particles when only the target analyte is present in the analyzed sample.

**Figure 2.**Time evolution of the expected value of the adsorbed particles numbers of two analytes, obtained by using the stochastic model of sensor response (solid lines), and the time dependence of the numbers of adsorbed particles according to the deterministic model of sensor response (dashed lines).

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**MDPI and ACS Style**

Jokić, I.; Djurić, Z.; Radulović, K.; Frantlović, M. Analysis of Stochastic Time Response of Microfluidic Biosensors in the Case of Competitive Adsorption of Two Analytes. *Proceedings* **2018**, *2*, 991.
https://doi.org/10.3390/proceedings2130991

**AMA Style**

Jokić I, Djurić Z, Radulović K, Frantlović M. Analysis of Stochastic Time Response of Microfluidic Biosensors in the Case of Competitive Adsorption of Two Analytes. *Proceedings*. 2018; 2(13):991.
https://doi.org/10.3390/proceedings2130991

**Chicago/Turabian Style**

Jokić, Ivana, Zoran Djurić, Katarina Radulović, and Miloš Frantlović. 2018. "Analysis of Stochastic Time Response of Microfluidic Biosensors in the Case of Competitive Adsorption of Two Analytes" *Proceedings* 2, no. 13: 991.
https://doi.org/10.3390/proceedings2130991