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The optimization-based quantitative determination of multianalyte concentrations from biased biosensor responses is investigated under internal and external diffusion-limited conditions. A computational model of a biocatalytic amperometric biosensor utilizing a mono-enzyme-catalyzed (nonspecific) competitive conversion of two substrates was used to generate pseudo-experimental responses to mixtures of compounds. The influence of possible perturbations of the biosensor signal, due to a white noise- and temperature-induced trend, on the precision of the concentration determination has been investigated for different configurations of the biosensor operation. The optimization method was found to be suitable and accurate enough for the quantitative determination of the concentrations of the compounds from a given biosensor transient response. The computational experiments showed a complex dependence of the precision of the concentration estimation on the relative thickness of the outer diffusion layer, as well as on whether the biosensor operates under diffusion- or kinetics-limited conditions. When the biosensor response is affected by the induced exponential trend, the duration of the biosensor action can be optimized for increasing the accuracy of the quantitative analysis.

Amperometric biosensors are analytical devices that measure the changes in the output current on the working electrode, due to the direct oxidation or reduction of the products of biochemical reactions [

Multianalytes have been successfully analyzed by biosensors using different multivariate approaches (e.g., partial least squares regression, principal component analysis) [

The multianalytes determination becomes even more complex if the biosensors response is perturbed by noise, e.g., white noise, sinusoidal power electrical noise or if the biosensor response is biased, e.g., by temperature change [

Recently, the problem of multianalytes determination by reverse problem solving has been explicitly formulated and successfully applied to optimize the calculation of multianalyte concentration using a response of a biocatalytic amperometric biosensor utilizing a mono-enzyme-catalyzed (nonspecific) conversion of multi-substrates [

Real biosensors contain as an outer membrane a thin layer of porous or perforated polyvinyl alcohol, polyurethane, cellulose, latex or other material [

The aim of this work is to investigate the effect of the external diffusion limitation on the precision of the determination of the multianalyte concentrations from the biased response of a biocatalytic amperometric biosensor utilizing a mono-enzyme-catalyzed multi-substrate conversion. The influence of the white noise-, as well as temperature-induced trend one the precision of multianalyte determination is also investigated. The investigation was carried out at very different catalytic conditions, thicknesses of the diffusion layer and types of signal noise.

Since a comprehensive analysis of a chemometric technique to be used for quantitative analysis usually requires a lot of input data, the biosensor responses to mixtures of compounds were simulated. On the other hand, computer simulation is usually much cheaper and faster than real experiments. Pseudo-experimental responses to mixtures of two compounds were numerically simulated using a computational model of the amperometric biosensor utilizing a mono-enzyme-catalyzed (nonspecific) conversion of two substrates [

The numerical simulation is based on a two compartment mathematical model involving coupled reaction-diffusion equations [

The numerical experiments showed that the precision of the concentration estimation significantly depends on the relative thickness (Biot number) of the outer diffusion layer. The accuracy of concentration estimation also depends on the biosensor response being under either the diffusion or the enzyme kinetics control,

We consider a mono-enzyme dual biosensor (two-substrates) utilizing the Michaelis-Menten kinetics [_{1} and S_{2} are the substrates to be determined, ES_{1}, ES_{2} stand for the enzyme-substrate complexes, P_{1}, P_{2} are the reaction products and _{+1}, _{−1}, _{+2}, _{−2}, _{3}, _{4} are the kinetic constants.

When two substrates (S_{1} and S_{2}) react with a single enzyme, E, without the formation of any two substrate complex, and the substrates do not combine directly with each other, then each substrate in a mixture of S_{1} and S_{2} acts as a competitive inhibitor of the others [_{1} and S_{2}, and 20

The reactions in the network given by

The amperometric biosensor is treated as an electrode, and a relatively thin layer of an enzyme (enzyme membrane) is applied onto the electrode surface. The biosensor model involves three regions: the enzyme layer, where the biochemical reactions (

Assuming a symmetrical geometry of the electrode and a homogeneous distribution of the immobilized enzyme in the enzyme layer of a uniform thickness, the mathematical model of the biosensor action can be defined in a one-dimensional-in-space domain [

Coupling the enzyme-catalyzed reactions given by _{i,e}(_{i,e}(_{i}_{i}_{i}_{i}_{Si,e}_{Pi,e}_{1} = _{3}_{0}, _{2} = _{4}_{0}, _{1} = (_{−1} + _{3})/_{+1}, _{2} = (_{−2} + _{4})/_{+2}, and _{0} is the total concentration of the enzyme,

Outside the enzyme layer, only the mass transport by diffusion of the substrates and products takes place,
_{i,b}_{i,b}_{i}_{i}_{Si,b}_{Pi,b}

The biosensor operation starts when both substrates (S_{1} and S_{2}) appear in the bulk solution (_{i}_{,0} is the concentration of the substrate, S_{i}

At the electrode surface (_{1} and P_{2}) are permanently reduced to zero, while for the non-ionized substrates, their fluxes are assumed to be zero (

Assuming a well-stirred buffer solution leads to the constant thickness of the diffusion layer, as well as the constant concentration above that layer during the biosensor operation (

On the boundary between two adjacent layers, the merge conditions are defined for the substrates, as well as the products (

The diffusion layer (

The measured anodic or cathodic current is usually assumed as the response of an amperometric biosensor. When modeling amperometric biosensors, due to the direct proportionality of the current to the area of the electrode surface, the current is often normalized with that area [_{1} and _{2} are the numbers of electrons involved in charge transfers in the corresponding electrochemical reactions at the electrode surface, _{∞} is the density of the steady-state biosensor current and

The diffusion module or Damköhler number essentially compares the rate of the enzyme reaction (_{i}_{i}_{Si,e}^{2}) [

where

If the diffusion module is less then unity, then enzyme kinetics (reaction rate) controls the biosensor response. The response is controlled or limited by diffusion when the module is greater than unity [

The Biot number is another dimensionless parameter widely used to indicate the internal mass transfer resistance to the external one [_{1} and _{2} are two Biot numbers corresponding to the diffusion of two substrates.

In real-life experiments, the reaction rate can be affected by the temperature change that induces a trend of the output signal of the sensor [_{T}

Assuming that the temperature, _{0} + _{a}_{0} = 298 K, and _{T}

Besides the signal trend, the biosensor response can be also affected by an unpredictable noise [_{N}_{TN}

Due to the nonlinearity of the governing

Explicit difference schemes have a convenient algorithm of the calculation and are simple for programming [_{1},_{2})^{2}), where _{1} = _{2} = _{1}, _{2}) ≤ (_{1} + _{2})/2 [

The mathematical, as well as the corresponding computational models of the biosensor were validated using known analytical solutions for mono-enzyme single substrate amperometric biosensors [_{i,e}_{i}_{,0} ≪ _{i}_{i}_{i}_{i}

For validating the model in the opposite case of the substrate concentrations, a high concentration, _{i}_{,0}, of the substrate, S_{i}_{1,0} ≫ _{1}, _{2,0} = 0 and _{2,0} ≫ _{2}, _{1,0} = 0. In both these cases of the extreme concentrations of the substrates, the initial boundary value problem,

The calculated values of the reaction term _{1} and S_{2}, and 20

The following values of the model parameters were constant in all the numerical experiments:
^{−6} cm^{2}/s, was used, which is typical for the enzyme membranes of biosensors and is two times less than that in the buffer solution [

In order to investigate the impact of the external diffusion limitation on the precision of the determination of the multianalyte concentrations and to avoid results dependent on the predefined values _{Si,b}_{Pi,b}_{1} and _{2}, so that the response would be in both enzyme kinetics and diffusion limitations.

The numerical solutions of the model, _{i}_{i}_{,0},

The rate of the trend is characterized by the activation energy, _{a}_{a}

The evolution of the typical biosensor responses is illustrated in _{1} = 0.5 μM/s for the first substrate and _{2} = 10 V_{1} = 5 μM/s for the second one; the concentration of the first substrate was _{1,0} = 3.2_{1} = 0.32 mM and _{2,0} = 4 _{1,0}= 12.8_{2} = 1.28 mM of the second substrate, and the thickness of the outer diffusion layer was _{1} = _{2} = 0.4 and the diffusion modules:

We consider the quantification of mixtures of two substrates from the response of the mono-enzyme biosensor. Similar problems for simpler models have been already considered in [

Let _{1},…, _{n}_{1}, …, _{n}_{1}, _{n}_{1}, _{2}) stands for the concentrations of the substrates, S_{1} and S_{2}, which are the subject to be evaluated, and _{i}_{i}

The concentrations c = (_{1}, _{2}) can be evaluated by tuning the values of the concentrations with respect to the conformity of theoretical measurements, _{1}, _{2}).

When the biosensor output is affected by the exponential trend _{−}, _{+}].

The optimization problem defined in _{a}

The optimization problems stated above are difficult to solve, due to mathematically unprovable convexity and uni-modality properties of the objective function, thus limiting the applicability of local optimization algorithms, and are highly computational resource-consuming to find a single value of the objective function. The approach to apply a surrogate objective function has been proposed in [_{i}_{i}_{i}_{1}, _{2}) for all

In the optimization problem given by _{1}, …, _{n}_{1}, …, _{n}_{1} and _{2}, to be determined, and the feasible region, _{n}

The feasible region for the concentrations _{1}, _{2}) has been chosen to be ^{2}, assuming that the concentrations are dimensionless and normalized with respect to the corresponding Michaelis constant as follows: _{i}_{i}_{,0}/_{i}_{i}_{,0} > _{i}

The entire domain, ^{2} = 9, 409 in total) comprise the discrete set, _{i}_{1}, _{2}) ∈ _{0} = 0; _{i}_{+1} − _{i}_{n}

The computation of a pseudo-experimental response _{1}, _{n}_{1}, _{2}) of the concentrations requires from around 15 to around 35 min (depending on the thickness of the diffusion layer and the reaction rate) using one core of an Intel Xeon X7550 (2 GHz) processor. Therefore, the computation of the responses for the whole set, _{1}, _{2}) of the concentrations within reasonable time.

Since the simulation of a single biosensor response requires from 15 up to 35 min when using a single core of an Intel Xeon X7550 processor, the simulation of 9, 409 such responses would require more than 2, 352 h, whereas the latter computations using 256 cores of Intel Xeon X7550 processor require less than 10 h. This indicates the almost linear speed-up of the computations or, in the other words, a reduction of the runtime by almost 256 times.

The obtained pseudo-experimental responses,

The optimization-based method for the evaluation of the concentrations has been implemented in a MATLAB environment using the Multi-Startstrategy with an efficient local minimization function,

The simulation of the biosensor responses is much more time-consuming than the optimization process; therefore the response simulator has been programmed in C++, including distributed memory parallel programming libraries.

The standard stopping criteria for the local search has been defined by a tolerance ^{−5} of the function value [^{5}.

The optimization-based method for the evaluation of the concentrations (see Section 3.2) has been used to investigate the precision of the evaluation of the concentrations, when the biosensor response follows the model, _{2}, for the second substrate is greater than the rate, _{1}, for the first one, _{2} = 10_{1} and

The quality of the quantitative analysis is influenced by whether the biosensor response is under the diffusion or the enzyme kinetics control, and the concentration estimation is usually more accurate for a substrate corresponding to a greater diffusion module than for another substrate corresponding to a lower diffusion module [_{1} and _{2}, were chosen, so that the biosensor response can be controlled by the enzyme kinetics for the first component (
_{1} = 0.5 μM/s, _{2} = 10 _{1} = 5 μM/s,

The precision of the evaluation has been determined by attempting to evaluate the set of 64 pairs _{1}, _{2}) of predefined concentrations for which the responses were simulated (see _{1}, _{2}) obtained by the optimization-based method. The precision of a single estimate of the concentration has been determined by the relative error,

The sensitivity of the estimates of the concentrations is illustrated by _{1} = 0.5 μM/s, _{2} = 5 μM/s assuming zero thickness (_{1}, _{2}) of the substrates were equal to (8, 8). As one can see in

The maximal time _{n}_{m}_{m}_{m}_{n}_{m}_{n}

In order to investigate the impact of the thickness, _{1} = _{2}) would vary in a wide range (from 0.4 up to ∞), assuming the thickness,

Since the concentrations from the noise-free biosensor responses can be evaluated very precisely [

The response time is an important characteristic of biosensors. In various applications, it is important to have the response time as short as possible [

Computational results showed that for all practical values of the thickness, _{m}_{n}_{m}_{i}_{i}_{n}_{m}_{n}

_{m}_{i}_{i}_{m}_{m}_{i}

Since the maximal enzymatic rate of the second substrate was notably (ten-fold) greater than of the first one, the second substrate has more impact on the biosensor response [_{2}, of the second substrate was evaluated notably more precisely than that of the first one in all the numerical experiments that were performed. Taking these considerations into account, bellow, the investigation is focused on the evaluation only of the concentration, _{1}, as less precisely determined.

Similar computational experiments were applied for investigating the impact of thickness

The maximal relative errors, _{1} and _{2}, of the concentrations, _{1} and _{2}, were less than 0.02 and 0.005, respectively. The most intractable situations occurred when the diffusion layer was the thickest (_{m}

_{1}, of the estimate, _{1}, of the substrate, S_{1}, _{m}_{1}, was calculated for different values of the Biot number,

One can see in _{m}_{m}_{m}_{m}

_{m}

One can see in _{1}, of the evaluation of the substrate concentration as a function of the Biot number can be of a different monotonicity. The error, _{1}, is a monotonous decreasing function of the Biot number, _{m}_{m}_{1}(

Only the dynamics of the biosensor current up to the steady state contains the full information on the dynamics of the biosensors operation. A shorter evolution of the biosensor current contains limited information used in the evaluation of the substrate concentration. The duration of the biosensor operation is especially important at the external diffusion limitation, _{1}, of the concentration evaluation appear in the case of a thick diffusion layer (small values of

Increasing the thickness of the external diffusion layer creates an additional diffusion limitation to the substrates, _{m}_{1}, of the concentration evaluation decreases with decreasing the Biot number,

In the numerical experiments discussed above, the values _{1} = 5 × 10^{−7} and _{2} = 5 × 10^{−6} M/s of the maximal enzymatic rates were chosen so that the biosensor response would be under mixed control, controlled by the enzyme kinetics for the first component (

the biosensor response to both substrates is controlled by the enzyme kinetics (_{1} = 5 × 10^{−8}, _{2} = 5 × 10^{−7}M/s,

the biosensor response to both substrates is controlled by the mass transport (_{1} = 5 × 10^{−6}, _{2} = 5 × 10^{−5} M/s,

The calculations showed that the difference between the evaluation precisions of different substrates changes with the changing of the maximal enzymatic rates, while the concentration, _{2}, of the second substrate, S_{2}, was evaluated notably more precisely in all the numerical experiments discussed above.

_{1} and _{2}, of the evaluation of the concentrations of both substrates, S_{1} and S_{2}, _{m}_{1} and _{2}, and a moderate value of the Biot number

As one can see in _{1}, of the concentration evaluation for the substrate, S_{1}, is noticeably greater than the error, _{2}, for the second substrate, S_{2}. This is especially noticeable for short durations of the biosensor operation, _{m}_{2}, becomes slightly greater than the error, _{1}, when long-term responses are used in the concentration evaluation.

It is known that when the response of a catalytic biosensor is considerably controlled by the mass transport, the steady-state current practically does not depend on the maximal enzymatic rate (total concentration of enzyme) [

The maximal enzymatic rate, _{1}, is actually a product of two parameters: the catalytic constant, _{3}, introduced in _{0}, of the enzyme, and correspondingly, _{2} is a product of _{0} and the constant, _{4}, introduced in _{1} = _{3}_{0}, _{2}_{4}_{0} [_{3} and _{4}, the maximal rates, _{1} and _{2}, as well as the diffusion modules,
_{0}, of the enzyme in the enzyme layer.

The optimization-based method of the quantitative analysis of the biosensor response proposed in [

The mathematical model,

If the biosensor signal is not affected by the temperature-induced trend, then the substrate concentrations are most accurately evaluated from the biosensor transient response recorded up to a steady state. Shortening the duration of the biosensor operation reduces the accuracy of the evaluation, especially in the case of a relatively thick outer diffusion layer (small values of the Biot number, β;

At different diffusion limitations and durations of the biosensor operation, the concentrations are more accurately evaluated when the response is not affected by a trend rather than affected by an induced exponential trend (

At different durations of the biosensor operation and rates of the induced trend, the concentrations of the substrates are more accurately evaluated when the biosensor response to the substrates is controlled by the enzyme kinetics rather than the response being controlled by the mass transport (

This research was funded by the European Social Fund under the Global Grant measure, project no. VP1-3.1-ŠMM-07-K-01-073/MTDS-110000-583.

Romas Baronas contributed to the computational modeling of the biosensor and to coordination of manuscript writing. Juozas Kulys was focused on the formulation of the general problem of white noise and temperature induced biocatalytical biosensors response drift and solving analyte concentration determination. Algirdas Lančinskas contributed to solving the optimization problem and conducted the computational experiments. Antanas Žilinskas contributed to the development of optimization-based method for assessment of the multianalyte concentrations. All authors contributed to the design and interpretation of the computational experiments, drafting and revision of the manuscript. All authors approved the final version of the article.

The authors declare no conflict of interest.

Principal structure of the biosensor.

The dynamics of the biosensor current affected by the exponential trends (_{1} = 0.5 μM/s, _{2} = 5 μM/s, _{1,0} = 0.32 mM, _{2,0} = 1.28 mM. The other parameters are as defined in

The set of the dimensionless concentrations, _{1} and _{2}, of the substrates, S_{1} and S_{2}, used in the investigation, as well as the noise-free biosensor responses to these concentrations simulated at and _{1} = 0.5 μM/s, _{2} = 5 μM/s and

Contour lines of the objective function given by _{1} = 0.5 μM/s, _{2} = 5 μM/s assuming zero (_{1}, _{2}) = (8, 8).

The impairment, Δ_{i}_{i}_{i}_{i}_{m}

The relative error, _{1}, of the evaluation of the concentration, _{1}, _{m}

The relative error, _{1}, of the evaluation of the concentration, _{1}, of the substrate, S_{1}, _{m}

The relative errors, _{1} and _{2}, of the evaluation of the concentrations, _{1} and _{2}_{1} and S_{2}, _{m}_{1} and _{2}, and the Biot number

The values of the parameters of the exponential trend

_{a} |
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Trend 1 | 6,000 | 3.33 × 10^{−3} |

Trend 2 | 24,000 | 3.33 × 10^{−3} |

Trend 3 | 24,000 | 6.66 × 10^{−3} |