# Stochastic Time Response and Ultimate Noise Performance of Adsorption-Based Microfluidic Biosensors

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## Abstract

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## 1. Introduction

## 2. Method—Mathematical Modeling of Biosensor Stochastic Time Response

_{a}and k

_{d}are the adsorption and desorption rate constants, respectively, C

_{S}is the analyte concentration in the immediate vicinity of the binding sites on the sensing surface of area A, N

_{m}is the number of binding sites on the surface, and k

_{m}is the mass transfer coefficient, introduced in TCM as k

_{m}= 1.467(D

^{2}v

_{m}/(L

_{s}h

_{c}))

^{1/3}[31] in order to characterize the transport of analyte particles by both convection and diffusion between the bulk solution and the surface binding sites (D is the diffusion coefficient of analyte particles, v

_{m}is the mean convection velocity, L

_{s}is the adsorption zone length, and h

_{c}is the sensor chamber height). According to TCM, all quantities are averaged across the sensing surface. The effective rates of the increase and decrease in the number of adsorbed particles, a(t) and d(t), respectively, do not depend explicitly on t, but on the instantaneous value of N, thus a(N) and d(N) in Equation (1).

_{RM}(N) and d

_{RM}(N) are actual adsorption and desorption rates.

_{N}(n, t), for the given initial state N(0) = n

_{0}(here, n

_{0}= 0 as t = 0 is assumed as the moment when the AD process starts on the sensing surface), is given by the master equation:

_{m}}, where N

_{m}is the total number of adsorption sites on the sensing surface), while A(n)dt and D(n)dt are the probabilities of transition from the state n to the state n + 1, and from the state n to the state n − 1 during the time interval dt→0, respectively. A(n) and D(n) are the probability of the increase in the number of adsorbed particles by 1 in unit time, and the probability of the decrease in N by 1 in unit time, respectively, when the current state is N = n. Equation (3) is valid for n = 0 if we define P

_{N}(−1,t) = 0 and A(−1) = 0, and it is also valid for n = N

_{m}assuming P

_{N}(N

_{m}+ 1,t) = 0 and D(N

_{m}+ 1) = 0 (D(0) = 0 due to the nature of the desorption process, and A(N

_{m}) = 0 because the sensing surface adsorption capacity is limited to N

_{m}).

^{2}, of the random number of adsorbed particles are obtained [37]:

_{a}C

_{s}(N

_{m}− n) = k

_{a}(C + k

_{d}n/(k

_{m}A))(N

_{m}− n)/(1 + k

_{a}(N

_{m}− n)/(k

_{m}A)) and D(n) = k

_{d}n. In this way, the expressions are obtained for the effective probabilities of the increase and decrease in the number of adsorbed particles per unit time, which combine the influences of the AD and MT processes. A(n) and D(n) depend on the current state N = n (which is a feature of birth–death processes). After representing the nonlinear transition rate as a Taylor series centered at the expected value, Equations (5) and (6) take the approximate form, which includes the first and the second moments:

^{2}(with the conditions <N> = 0 and σ

^{2}= 0 at the moment t = 0).

^{2}/dt = 0:

_{RM}(n) = k

_{a}C(N

_{m}− n) and D

_{RM}(n) = k

_{d}n. In that case, Equations (5) and (6) yield the system of exact equations:

_{RM}= 1/(k

_{d}+ k

_{a}C). The time-dependent SNR and its steady-state value in the case of neglected MT influence are obtained from Equations (11) and (15), respectively, by using <N>

_{RM}, σ

_{RM}, <N>

_{RM}

_{,e}, and σ

_{RM}

_{,e}instead of the corresponding parameters of the model that includes the MT effect.

## 3. Results and Discussion

#### 3.1. Analysis of Time Evolution of the Expected Value and Variance of the Number of Adsorbed Particles and Sensor Signal-to-Noise Ratio, Considering MT Influence

^{16}to 6·10

^{18}m

^{−3}). The curves shown as solid lines in the presented diagrams are obtained by using the stochastic model given by Equations (9) and (10), which is numerically solved. The diagrams enable the investigation of the kinetics of the stochastic sensor response, considering MT effects. The AD process parameters are k

_{a}= 1.33·10

^{−19}m

^{3}s

^{−1}and k

_{d}= 0.08 s

^{−1}, there are N

_{m}= 3·10

^{6}adsorption sites on the sensing surface of area A = 10

^{−9}m

^{2}, and the mass transfer coefficients are k

_{m}

_{1}= 2·10

^{−2}ms

^{−1}for Figure 2a and k

_{m}

_{2}= 2·10

^{−5}ms

^{−1}for Figure 2b. All the parameter values are very close to those used in [31], for which the TCM applicability has been demonstrated in the same work.

_{m}. The slower mass transfer process (low k

_{m}) prolongs the transient period of the time response at all concentrations, while the influence on the equilibrium expected value is not noticeable for the given set of parameter values. These conclusions are in accordance with those corresponding to the response kinetics described by the deterministic model (Equation (1)) [28,31].

_{m}> k

_{m}

_{1}match those shown in Figure 2a, which means that for k

_{m}= k

_{m}

_{1}, mass transfer is already sufficiently fast to be of a negligible influence on the response kinetics. This explains the matching of curves (solid lines in Figure 2a), obtained by using the stochastic model that takes into account the coupling of stochastic AD and MT processes characterized by the coefficient k

_{m}

_{1}, with those obtained by using the stochastic model that neglects the influence of MT (dashed lines in Figure 2a, entirely covered by solid lines). Indeed, the expressions given in Section 2 show that, for a sufficiently high k

_{m}, transition rates according to the model that takes into account MT, A(n) and D(n), become equal to k

_{a}C(N

_{m}− n) and k

_{d}n, respectively, which are the well-known expressions for transition rates A

_{RM}(n) and D

_{RM}(n), respectively, valid when MT effects are negligible. This means that, for a sufficiently high k

_{m}, the derived stochastic model, given by Equations (9) and (10), reduces to the model presented by Equations (18) and (19), i.e., the former model is a superset of the latter. Therefore, the model that takes into account the coupling of AD and MT processes covers the cases of both the pronounced and negligible MT effects on the stochastic response, and it is in this example applied for the research of microfluidic sensor kinetics both in the case when the MT influence is significant, i.e., in the mass transfer-limited regime (as shown in Figure 2b), and in the case when the MT influence is negligible, i.e., in the regime of rapid mixing or the adsorption-limited kinetics (as shown in Figure 2a).

_{m}

_{1}= 2·10

^{−2}ms

^{−1}, and it corresponds to the expected value given in Figure 2a, while the diagram in Figure 3b is for k

_{m}

_{2}= 2·10

^{−5}ms

^{−1}, and it constitutes a pair with the diagram in Figure 2b. The solid line curves in Figure 3a,b are obtained by using the theoretical model that takes into account AD and MT processes (Equations (9) and (10)), while the dashed lines in Figure 3a represent the variances determined according to the model that neglects the MT influence (Equations (18) and (19)).

^{17}m

^{−3}and lower, σ

^{2}(t) is a monotonically increasing function. As the concentration increases to 6·10

^{17}m

^{−3}, the transient regime duration decreases, and the equilibrium variance value increases. With the further increase in the concentration, the dependence σ

^{2}(t) has an increasingly prominent peak, the duration of the transient regime continues to decrease, while the steady-state variance value decreases.

^{2}(t) at lower concentrations (noticeable even at C = 4.2·10

^{17}m

^{−3}). As the concentration increases, the transient regime duration decreases, and the steady-state variance value first increases and then decreases with the concentration, in the same way as in the case of high k

_{m}(Figure 3a). The peak becomes increasingly pronounced with the concentration beyond 4.2·10

^{17}m

^{−3}, and the maximal variance (corresponding to the peak) noticeably decreases. These conclusions stemming from our model, regarding the time-dependent variance when the MT influence is pronounced, are in accordance with the results of the stochastic computer simulation, which is based on the model that takes into account coupled AD and diffusion (called “coupled hybridization-diffusion process” in the mentioned reference) in nanowire biosensors and presented in [18].

_{m}

_{1}is very close to the specific value above which the MT influence on the variance becomes negligible. For every k

_{m}value greater than that specific value, the model that takes into account MT yields the same curve σ

^{2}(t) as σ

_{RM}

^{2}(t) for a given C. When the MT influence is negligible, the analysis of Equation (19) shows that the function σ

_{RM}

^{2}(t) has a maximum (peak) at concentrations C > k

_{d}/k

_{a}= 6·10

^{17}m

^{−3}, and that this maximum does not change as the concentration increases further (the peak height is independent of C and equal to N

_{m}/4), as can be seen in Figure 3a. At the moment when the variance is at its peak value, the expected value <N>

_{RM}equals N

_{m}/2 (which is obtained from Equation (18)). σ

_{RM}

^{2}(t) is a monotonically increasing function for C < k

_{d}/k

_{a}(then <N>

_{RM}<N

_{m}/2), and σ

_{RM}

^{2}(t) ≈ <N>

_{RM}(t) is valid for C << k

_{d}/k

_{a}. If the measurement of the signal fluctuation is used as an analytical tool in biosensing, the position and the value of the variance maximum can provide information in addition to those obtained by the noise analysis in the steady state. For example, in the case of negligible mass transfer influence, the value of the variance maximum can be used for the estimation of the number of surface binding sites N

_{m}, which is a parameter important for the estimation and optimization of the sensor performance. In addition, the existence of the variance overshoot indicates that C > k

_{d}/k

_{a}.

_{m}

_{1}), and in the case when the mass transfer influence is pronounced (k

_{m}

_{2}), respectively, for different analyte concentrations (the values of all parameters are given at the beginning of Section 3.1, and they are the same as for Figure 2a,b and Figure 3a,b). The curves corresponding to the cases when the MT influence is negligible (obtained according to the model that neglects MT, given by Equations (18) and (19)) are so close to those shown in Figure 4a (for k

_{m}

_{1}= 2·10

^{−2}ms

^{−1}) that the difference between them is not noticeable in the diagram of that scale. The diagrams show that the SNR decreases with the decrease in C, for every t, both for rapid and slow mass transfer. Mass transfer increases the time needed for SNR to achieve its maximum value (corresponding to the steady state) at a given concentration. In addition, slow MT decreases the SNR for the given analyte concentration. Therefore, it depends on the value of k

_{m}whether or not it is possible to reach the required SNR for reliable detection and quantification of an analyte by using a given sensor with a given set of parameter values.

#### 3.2. Analysis of MT Influence on the Sensor Stochastic Response and Noise Performance in Steady State

_{m}. The difference between the corresponding quantities determined by using the two models can thus be used as a measure of MT influence. The MT with the coefficient k

_{m}= 2·10

^{−5}ms

^{−1}is assumed, the adsorption sites surface density is n

_{m}= N

_{m}/A = 3·10

^{15}m

^{−2}for all analyzed sensors, and the remaining parameter values are those given at the beginning of Section 3.1, unless otherwise noted.

^{−12}to 10

^{−9}m

^{2}, for different concentrations of the target protein. The curves obtained according to the two stochastic models match, which means that the influence of MT on the steady-state expected value is negligible for the given set of parameter values. The expressions for the steady-state expected value according to the two models, given by Equations (12) and (20), yield the ratio:

_{RM}

_{,e}/<N>

_{e}≈ 1 is valid):

_{m}corresponds to the case when k

_{m}

_{,ev}has the highest value, i.e., at the lowest analyte concentration, and in sensors of the smallest sensing surface area from the considered range. For the sets of parameter values used in our analysis (Figure 5a), the maximal k

_{m}

_{,ev}equals 1.2·10

^{−7}ms

^{−1}, so the most stringent condition for the expected values obtained according to the two models to be approximately equal is k

_{m}>> 1.2·10

^{−7}ms

^{−1}, which is fulfilled for k

_{m}= 2·10

^{−5}ms

^{−1}. This explains the matching of the curves obtained by using the two models (Figure 5a). The condition (23) is satisfied in a wide range of parameter values. However, in sensors with extremely small sensing surfaces (such as nanowire or carbon nanotube mechanical or electrical sensors), as well as in detection of particles present in ultra-low concentrations, the value of k

_{m}

_{,ev}can be such that the condition given by Equation (23) is not satisfied, which implies that MT could influence the expected value of the sensor stochastic response by decreasing it. This conclusion is in accordance with the result of a computer simulation performed for a nanowire biosensor in [29], which showed the decrease in the expected value of the number of adsorbed particles in the case of binding influenced by diffusion. This deserves further investigation by using a model particularly considering nanoscale sensors.

_{m}= 2·10

^{−5}ms

^{−1}causes a significant increase in the steady-state variance at a given concentration and sensing surface area. The condition for MT to be of negligible influence on the variance is obtained from σ

^{2}

_{e}≈ σ

^{2}

_{RM,e}. It can be formulated through the ratio:

_{m}is obtained, under which the ratio approximately equals 1:

^{2}

_{e}/σ

^{2}

_{RM,e}) does not depend on A. Indeed, the analysis of the expression σ

^{2}

_{e}/σ

^{2}

_{RM,e}shows that for A >> k

_{a}/k

_{m}= 6.65·10

^{−15}m

^{2}(satisfied for all A values within the considered range), the following is valid:

_{m}at which the MT influence on the variance is insignificant, i.e., the applicability condition for the simpler stochastic model:

^{18}m

^{−3}, but even then, k

_{m}

_{,var}≈ 3.6·10

^{−5}ms

^{−1}. Due to that, the variances determined according to the two models differ significantly when the MT coefficient equals 2·10

^{−5}ms

^{−1}, for every C value from the considered range.

_{a}/k

_{m}= 6.65·10

^{−15}m

^{2}is valid). By using the model that takes into account the MT influence, and for k

_{m}that satisfies the condition (27), the curves are obtained that match those shown as dashed lines in Figure 5c.

_{m}influences the maximal achievable SNR value of a sensor with a given sensing surface area. The diagram shown in Figure 5c can be used to determine whether or not it is possible to achieve an SNR value required to reliably detect and quantify the target substance concentration using a sensor of a given surface area. It can be seen that the steady-state SNR of a sensor with the sensing surface area of 10

^{−12}m

^{2}, protein concentration of 6·10

^{16}m

^{−3}, and MT coefficient of 2·10

^{−5}ms

^{−1}approximately equals 3, which is the minimal value needed for protein detection [40,41]. The same sensor in the case of negligible MT influence has an SNR of almost 20, so it satisfies the more stringent condition (SNR ≥ 10 [40]) for reliable quantification of the concentration. A sensor with a sensing surface area of 10

^{−11}m

^{2}in the case of k

_{m}= 2·10

^{−5}ms

^{−1}has an SNR higher than 10, so it satisfies the conditions for both analyte detection and quantification of the concentration even when the MT influence is pronounced. The same diagram enables the determination of the minimal detectable and measurable concentrations (i.e., the fundamental detection and quantification limits, as they are determined by the fundamental noise) of the given sensor, for the required SNR value (e.g., in [42], SNR = 1 was used for the estimation of the minimum detectable change in the measured quantity in a graphene ISFET, as the theoretical limit of performance determined by intrinsic noise; in [43], SNR = 1 was also used for the determination of the detection threshold in silicon nanowire sensors). These results show that the fundamental detection and quantification limits depend on the MT rate.

^{−9}m

^{2}, and they correspond to the steady-state values of time-dependent variances and SNRs shown in Figure 3 and Figure 4, respectively. For the remaining surface areas considered in Section 3.2 (from 10

^{−12}to 10

^{−10}m

^{2}), the conclusions will be the same as those obtained based on Figure 6b about the MT-influenced change in the variance and SNR (expressed through the ratios σ

^{2}

_{e}/σ

^{2}

_{RM,e}and SNR

_{e}/SNR

_{RM,e}) in the considered concentration range. This is because the analysis given in the comment for Figure 5b,c showed that the magnitude of the MT influence on these two quantities does not depend on the active surface area when A >> k

_{a}/k

_{m}= 6.65·10

^{−15}m

^{2}(then, the ratios σ

^{2}

_{e}/σ

^{2}

_{RM,e}(Equation (26)) and SNR

_{e}/SNR

_{RM,e}do not depend on A). In addition, as σ

^{2}

_{RM,e}is proportional to A, and it can be shown that for surface areas A >> k

_{a}/k

_{m}= 6.65·10

^{−15}m

^{2}, σ

^{2}

_{e}is also proportional to A, all conclusions about the dependences of σ

^{2}

_{e}and σ

^{2}

_{RM,e}on C, based on Figure 6a, will be valid for all sensing surface areas in the range from 10

^{−12}to 10

^{−9}m

^{2}. For a similar reason (SNR

_{e}and SNR

_{RM,e}are proportional to A

^{1/2}), conclusions based on Figure 6a about the influence of both MT and target substance concentration on the change in SNR of a sensor with the sensing surface area of 10

^{−9}m

^{2}are valid for other sensors of different sensing surface areas from the mentioned range.

_{max}

_{,RM}= k

_{d}/k

_{a}= 6·10

^{17}m

^{−3}. Starting from the expression for σ

^{2}

_{e}(Equation (13)), which is simplified under the condition A >> k

_{a}/k

_{m}, it can be analytically shown that the variance influenced by MT has the maximum at the concentration C

_{max}≥ C

_{max,RM}. Thus, due to the influence of MT, the AD noise maximum moves toward lower concentrations of the target substance.

^{2}

_{e}/σ

^{2}

_{RM,e}, given by Equation (26), asymptotically approaches the maximum value 1 + k

_{a}n

_{m}/k

_{m}≈21, while, when C increases, the ratio of variances approaches 1 (i.e., for given k

_{m}, the variances according to the two models are approximately equal at a sufficiently high concentration).

_{e}(C) and SNR

_{RM,e}(C), shown in Figure 6a, increase monotonically as the concentration increases. The influence of MT on the decrease in the sensor’s steady-state SNR is concentration-dependent. This can be quantitatively analyzed based on the diagram shown in Figure 6b. MT causes the greatest decrease in the sensor’s SNR at low concentrations. With the decrease in concentration, the ratio SNR

_{e}/SNR

_{RM,e}asymptotically approaches its minimum.

_{e}(C) obtained by using the derived analytical expression and shown in Figure 6a is in accordance with that obtained in [30] by computer simulation for the case of diffusion-influenced binding. Diagrams of this kind (Figure 6a) enable the determination of the concentration detection and quantification limits for a given sensor and given experimental conditions, as the values of C at which the SNR has the minimal required values for reliable analyte detection and quantification, respectively.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Adsorption-based microfluidic sensor: schematic representation of the typical system geometry with designations of dimensions and coordinate axes. The magnified partial cross-section of the microfluidic reaction chamber in the sensing surface zone is given as an illustration of the two-compartment model approximation of the spatially and time-dependent target substance concentration, affected by coupled adsorption–desorption and mass transfer processes of target analyte particles.

**Figure 2.**The expected value of the number of adsorbed particles in time, reflecting the kinetics of a biosensor stochastic response for different concentrations of the target protein in the analyzed sample: (

**a**) The case of negligible MT influence—the curves <N> (shown by solid lines) according to the model that considers MT for k

_{m}

_{1}= 2·10

^{−2}ms

^{−1}show the overlapping with the curves predicted by the model that neglects MT, <N>

_{RM}(dashed lines, entirely covered by solid lines). (

**b**) The case of MT influenced kinetics (k

_{m}

_{2}= 2·10

^{−5}ms

^{−1}).

**Figure 3.**Variance of the number of adsorbed particles, revealing the behavior of the sensor response variance, i.e., sensor’s AD noise during time, for different MT coefficient values: (

**a**) k

_{m}

_{1}= 2·10

^{−2}ms

^{−1}(solid lines correspond to the stochastic model that takes into account the coupling of AD and MT processes, dashed lines correspond to the model that neglects the MT influence); (

**b**) k

_{m}

_{2}= 2·10

^{−5}ms

^{−1}(according to the model that considers the combined effects of AD and MT).

**Figure 4.**Time dependence of the sensor signal-to-noise ratio according to the model that considers the combined AD and MT effects (solid lines), for two different values of the MT coefficient: (

**a**) k

_{m}

_{1}= 2·10

^{−2}ms

^{−1}; (

**b**) k

_{m}

_{2}= 2·10

^{−5}ms

^{−1}. The curves obtained according to the model that neglects MT (dashed lines) match those predicted by the model that takes into account the coupling of AD and MT processes for k

_{m}

_{1}(solid lines), as shown in (

**a**).

**Figure 5.**Expected value (

**a**) and variance (

**b**) of the number of adsorbed particles in the steady state, and the steady-state signal-to-noise ratio (

**c**), as a function of the sensing surface area, for different concentrations of the target protein, according to the stochastic model that takes into account the combined effect of AD and mass transfer processes (solid lines), and according to the stochastic model that neglects mass transfer (dashed lines).

**Figure 6.**(

**a**) Dependence of the variance of the number of adsorbed particles and sensor’s SNR (A = 10

^{−9}m

^{2}) in the steady state on the target substance concentration, according to the stochastic model that takes into account MT (solid lines), and the model that neglects it (dashed lines). (

**b**) Ratios of steady-state variances (σ

^{2}

_{e}/σ

^{2}

_{RM,e}) and SNRs (SNR

_{e}/SNR

_{RM,e}) according to the two models, depending on the concentration.

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**MDPI and ACS Style**

Jokić, I.; Djurić, Z.; Radulović, K.; Frantlović, M.; Milovanović, G.V.; Krstajić, P.M.
Stochastic Time Response and Ultimate Noise Performance of Adsorption-Based Microfluidic Biosensors. *Biosensors* **2021**, *11*, 194.
https://doi.org/10.3390/bios11060194

**AMA Style**

Jokić I, Djurić Z, Radulović K, Frantlović M, Milovanović GV, Krstajić PM.
Stochastic Time Response and Ultimate Noise Performance of Adsorption-Based Microfluidic Biosensors. *Biosensors*. 2021; 11(6):194.
https://doi.org/10.3390/bios11060194

**Chicago/Turabian Style**

Jokić, Ivana, Zoran Djurić, Katarina Radulović, Miloš Frantlović, Gradimir V. Milovanović, and Predrag M. Krstajić.
2021. "Stochastic Time Response and Ultimate Noise Performance of Adsorption-Based Microfluidic Biosensors" *Biosensors* 11, no. 6: 194.
https://doi.org/10.3390/bios11060194