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A mathematical model of biosensors acting in a trigger mode has been developed. One type of the biosensors utilized a trigger enzymatic reaction followed by the cyclic enzymatic and electrochemical conversion of the product (CCE scheme). Other biosensors used the enzymatic trigger reaction followed by the electrochemical and enzymatic product cyclic conversion (CEC scheme). The models were based on diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reactions. The digital simulation was carried out using the finite difference technique. The influence of the substrate concentration, the maximal enzymatic rate as well as the membrane thickness on the biosensor response was investigated. The numerical experiments demonstrated a significant gain (up to dozens of times) in biosensor sensitivity when the biosensor response was under diffusion control. In the case of significant signal amplification, the response time with triggering was up to several times longer than that of the biosensor without triggering.

The chemical amplification in analysis was reviewed almost 25 years ago [

The substrate cyclic conversion by conjugating the enzymatic reaction with chemical or electrochemical process was utilized in a single enzyme membrane [

If a biosensor contains an enzyme that starts analyte conversion followed by cyclic product conversion, the scheme of the biosensor action can be called a “triggering”. An example of this type of conversion is the amperometric detection of alkaline phosphatase based on hydroquinone recycling [

The triggering of the consecutive substrate conversion can also be realized by enzymatic conversion of the substrate (trigger reaction) followed by the second enzymatic reaction and electrochemical conversion. This scheme can be abbreviated as CCE. The scheme may be realized, for example, by using peroxidase and glucose dehydrogenase. The peroxidase produces an oxidized product that is reduced by GDH, thus realizing the cyclic conversion of the product. The goal of this investigation is to propose a model allowing computer simulation of the biosensor response utilising both schemes. The model developed is based on non-stationary diffusion equations [

A biosensor is considered as an enzyme electrode, containing a membrane with immobilised enzymes applied onto the surface of the electrochemical transducer. We assume the symmetrical geometry of the electrode and homogeneous distribution of immobilised enzymes in the enzyme membrane.

In the CEC scheme, the substrate (S) is enzymatically (E_{1}) converted to the product (P_{1}) followed by the electrochemical conversion of the product (P_{1}) to another product (P_{2}) that, in turn, is enzymatically (E_{2}) converted back to P_{1}:

Coupling the enzyme-catalysed reactions (_{i}(_{i}, respectively, _{i} is the maximal enzymatic rate, _{i} is the Michaelis constant, _{S} and _{Pi} are the diffusion coefficients,

Let _{0} is the concentration of substrate in the bulk solution.

The electrode potential is chosen to keep the zero concentration of the product P_{1} at the electrode surface. The rate of the product P_{2} generation at the electrode is proportional to the rate of conversion of the product P_{1}. When the substrate is well-stirred outside the membrane, the diffusion layer remains at a constant thickness (0 <

The biosensor current depends upon the flux of the product P_{1} at the electrode surface, i.e. at the border _{CEC}(_{1} of the product P_{1} at the surface of the electrode
_{e} is the number of electrons involved in a charge transfer at the electrode surface, and

We assume, that system (_{CEC} is the steady-state biosensor current.

In the CCE scheme, the substrate (S) is enzymatically (E_{1}) transformed to the product (P_{1}) followed by the enzymatic (E_{2}) conversion of the product P_{1} to another product P_{2} that, in turns, is electrochemically converted back to P_{1}:

If the thickness of enzyme membrane is

When the biosensor acts in the CCE mode, the electrode potential is chosen to keep zero concentration of the product P_{2} at the electrode surface. The rate of the product P_{1} generation at the electrode is proportional to the rate of conversion of the product P_{2}. Consequently, the boundary conditions (

The density _{CCE}(_{2} at the surface of the electrode:

When the system (_{CCE} of the biosensor acting in CCE mode

To compare the responses of trigger and normal biosensors, the action of the CE biosensor was analysed. In accordance to the CE scheme, the substrate (S) is enzymatically (E_{1}) converted to the product (P_{1}) followed by the electrochemical product (P_{1}) conversion to another product (P_{2}):

The mathematical model of a biosensor acting in CE mode can be derived from the model (_{2}, i.e. _{2} = 0. If _{CE}(_{CE} by (

To compare the amplified biosensor response with the response without amplification, we define the gain of the sensitivity as the ratio of the steady-state current of the trigger biosensor to the steady-state current of a corresponding CE biosensor
_{CEC}(_{1}, _{2}) and _{CCE}(_{1}, _{2}) are the steady-state currents of the trigger biosensors acting in CEC and CCE mode, respectively, at the maximal activity _{i} of an enzyme E_{i}, _{CE}(_{1}) is the steady-state current of the corresponding CE biosensor measured at the maximal enzymatic rate _{1} of an enzyme E_{1}, and _{CE}(_{1}) = _{CEC}(_{1}, 0).

Definite problems arise when solving analytically the non-linear partial differential equations with complex boundary conditions [_{2} = 0.

The finite difference technique [

An explicit scheme is easier to program, however, the implicit one is more efficient [

Due to the boundary conditions (^{−3}

In digital simulation, the biosensor steady-state time was defined as the time when the absolute current slope value falls below a given small value normalised with the current value. In other words, the time needed to achieve a given dimensionless decay rate

Consequently, the steady-state biosensor currents _{CEC} and _{CCE} were taken as the current at the biosensor response time _{CCE} and _{CCE}, respectively, _{CEC} ≈ _{CEC}(_{CEC}), _{CCE} ≈ _{CCE}(_{CCE}). In calculations, we used ^{−5}.

The mathematical models as well as the numerical solutions of the models were evaluated for different values of the maximal enzymatic rates _{1} and _{2}, substrate concentration _{0}, and thickness

In _{1} = _{2} = 100 nmol/(cm^{3}s), substrate concentration _{0} = 20 nmol/cm^{3} and membrane thickness _{M}, assuming _{M} = _{1} = _{2} = 5_{0}, _{0N} = 0.2:

The concentration profiles in

The steady-state current is similar for both types of biosensors, _{CEC} ≈ _{CEC}(123) ≈ 6.23 μA/cm^{2}, _{CCE} ≈ _{CCE}(124) ≈ 6.09 μA/cm^{2}. The time of steady-state is also approximately the same in both these cases. At the steady-state conditions, i.e. _{1}/_{2}/_{1}(_{2}(_{0} holds for all

The dependence of the steady-state current on the activity of both enzymes is shown in _{1} and _{2} varied from 10^{−10} to 10^{−6} mol/(cm^{3}s), the substrate concentration _{0} was 20 nmol/cm^{3}, _{0N} was 0.2 and membrane thickness _{CEC}(_{1}, _{2}) as well as _{CCE}(_{1}, _{2}) are monotonously increasing functions of both arguments: _{1} and _{2}.

In the case of CEC mode, an application of an active enzyme E_{2} (_{2} > 0) stimulates an increase of the biosensor current. In the case of _{2} = 0, the biosensor acting in CEC mode generates the current if only _{1} > 0. However, in the case of CCE mode, the appearance of an active enzyme E_{2} (_{2} > 0) is a critical factor for the biosensor current. _{CCE} = 0 if _{2} = 0 even if the activity of an enzyme E_{1} is very high (_{1} ≫ 0). Because of this, at low values of _{2}, the steady-state current _{CCE} increases very quickly with increase of _{2}. That effect is noted in _{CCE}(_{1}, _{2}) (_{CEC}(_{1}, _{2}) (

Consequently, when _{2} → 0 at any _{1} > 0, in the CEC mode: _{CCE}(_{1}, _{2}) → 0, while in another mode (CEC) of triggering: _{CEC}(_{1}, _{2}) → _{CEC}(_{1}, 0) = _{CE}(_{1}). On the other hand, _{CEC}(_{1}, _{2}) ≈ _{CCE}(_{1}, _{2}) at a high maximal enzymatic rate _{2}.

To investigate the effect of the amplification, _{CE}(_{1}) has been calculated at the same conditions as above. Having _{CEC}(_{1}, _{2}), _{CCE}(_{1}, _{2}) and _{CE}(_{1}), we calculated the gains _{CEC}(_{1}, _{2}) and _{CCE}(_{1}, _{2}). Results of calculations are depicted in _{2}. The increase is especially notable at high values of _{2}. The variation of _{1} on the response gain is slight by only. The gain varies from 18.0 to 19.1 at _{2} = 1 μmol/(cm^{3}s) in both action modes: CEC and CCE.

Comparing the gain in the CEC mode (_{2}. The gain _{CEC} starts to increase from about unity, while _{CCE} at low values of _{2} (_{2} < ≈ 1 nmol/(cm^{3}s)) is even less than unity. It means that in the case of low activity of enzyme E_{2}, the steady-state current of a biosensor is acting in the CCE mode even less than the steady-state current of a biosensor acting in the CE mode at the same conditions.

From the model of the CCE biosensor follows that _{2}(_{2} → 0. Consequently, _{CCE}(_{1}, _{2}) → 0 when _{2} → 0 at any _{1} > 0, while in the CEC mode: _{CEC}(_{1}, _{2}) → 1 when _{2} → 0. On the other hand, _{CEC}(_{1}, _{2}) ≈ _{CCE}(_{1}, _{2}) at a high maximal enzymatic rate _{2}, e.g. at _{2} = 1 μmol/(cm^{3}s).

To investigate the dependence of the signal gain on the substrate concentration _{0}, the response of biosensors varying _{0} from 10^{−10} to 10^{−4} mol/cm^{3} was simulated. Since the gain of trigger biosensors is significant only at a relatively high maximal enzymatic rate _{2} of enzyme E_{2} (_{2}: 10^{−6} and 10^{−7} mol/(cm^{3}s). We chose also two different values of the maximal enzymatic rate _{1} of enzyme E_{1}: 10^{−6} and 10^{−8} mol/(cm^{3}s). Since the influence of _{1} on the signal gain is not so significant as that of _{2}, the chosen two values of _{1} differ in two orders of magnitude while values of _{2} differ only in one. The results of calculations at the enzyme membrane thickness

As one can see in _{CEC} and _{CCE} is observed at high substrate concentrations only, _{0N} > 1. However, in a case of a higher value of _{2}, _{2} = 10^{−6} mol/(cm^{3}s), and a lower _{1}, _{1} = 10^{−8} mol/(cm^{3}s), no noticeable difference is observed between values of _{CEC} (curve 5 in _{CCE} (curve 6 in _{1}, _{1} = 10^{−6} mol/(cm^{3}s), and a ten times higher value of _{2}, _{2} = 10^{−7} mol/(cm^{3}s), curves 7 and 8.

_{2} to both signal gains: _{CEC} and _{CCE}. Such an importance is especially perceptible at low and moderate concentrations of substrate, _{0N} < 1. At _{0N} < 0.1 and _{2} = 1 μmol/(cm^{3}s) due to the amplification, the steady-state current increases up to about 18 times (_{CEC} ≈ _{CCE} ≈ 18). However, at the same _{0N} and ten times lower value of _{2}, the gain is about three times less, _{CEC} ≈ _{CCE} ≈ 5.7. Consequently, at low substrate concentrations, _{0N} < 0.1, and wide range of the maximal enzymatic rate _{1}, the tenfold reduce of _{2} reduces the signal gain about three times. This property is valid for both modes of triggering: CEC and CCE.

When increasing the substrate concentration, the signal gain starts to decrease when _{0N} becomes greater than unity (_{0} > _{1} = _{2}. However, the decrease is perceptible in cases of a high enzymatic rate _{1} only. At low activity of enzyme _{1} when _{1} = 1 nmol/(cm^{3}s), the gain varies less than 30% for both values of _{2}: 10 and 100 nmol/(cm^{3}s). Additional calculations showed, that at a less activity of enzyme _{1} when _{1} = 10^{−10} mol/(cm^{3}s), the gain practically does not vary changing the substrate concentration in the domain. Because of a very stable amplification at a wide range of substrate concentration, the usage of biosensors acting in a trigger mode is especially reasonable at a relatively low maximal enzymatic activity (rate _{1}) of enzyme E_{1} and a high activity (rate _{2}) of enzyme E_{2}. In the cases of relatively high maximal enzymatic activity _{1} the signal amplification is stable only for low concentrations of the substrate.

Additional calculations showed that the signal gain vanishes fast with the decrease of the enzymatic activity _{2} of enzyme E_{2}. For example, in the case of _{2} = 1 nmol/(cm^{3}s) the gain becomes less than 2 even at a low substrate concentration, _{CEC} ≈ 1.91, _{CCE} ≈ 1.3 at _{0N} = 0.01. This effect is also observed in _{2} reduces the signal gain about three times is valid at a wide range, also of _{2}.

A similar dependence of the signal gain on the substrate concentration was observed in the case of an amperometric enzyme electrode with immobilized laccase, in which a chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane [^{3}s) and the membrane thickness of 0.02 cm. For comparison of that gain with the gain achieved in the trigger mode, we calculate _{CEC} and _{CCE} for the enzyme membrane of thickness _{CEC} ≈ _{CCE} ≈ 34 at _{1} = _{2} = 1 μmol/cm^{3}s,

The steady-state current of membrane biosensors significantly depends on the thickness of the enzyme layer [_{1} of enzyme E_{1} and rate _{2} of enzyme E_{2} was simulated.

_{CEC} and _{CCE} versus the membrane thickness _{1} = 1 μmol/(cm^{3}s) and three values of the rate _{2}: 1, 10 and 100 nmol/(cm^{3}s). Comparing the gain _{CEC} with _{CCE}, one can notice valuable differences in behaviour of the signal gains. In the case of a CEC biosensor action, no notable amplification is observed in cases of a thin enzyme membrane (^{−3} cm). A more distant increase of the thickness causes an increase of the gain _{CEC}. The thickness at which _{CEC} starts to increase, depends on the maximal enzymatic rate _{2}.

The response of amperometric biosensors is known to be under mass-transport control if the diffusion modulus ^{2} is greater than unity, otherwise the enzyme kinetics controls the response:
_{max} is the maximal enzymatic rate and _{M} is the Michaelis constant. Since the diffusion coefficient _{S} and _{M} = _{1} = _{2} are constant in all our numerical experiments as defined in (_{2,} (_{σ} of the enzyme layer as a function of _{2} at which ^{2} = 1 has been introduced:

Comparing the value _{σ}(10^{−6}) ≈ 5.5×10^{−4} cm with the membrane thickness at which the gain _{CEC} starts to increase _{2} = 10^{−6} mol/(cm^{3}s), one can notice that the amplification becomes noticeable when the mass transport by diffusion starts to control the biosensor response. As one can see in _{2}: 10 and 100 nmol/(cm^{3}s). However, this is valid in the case of the biosensor acting in the CEC mode only. In the case of CCE mode, the gain _{CCE} increases notably with increase of the thickness _{CCE} is approximately a linear increasing function of _{CCE} > 1 if only _{σ}. As it was noticed above (see _{CCE} equals approximately to _{CEC}, _{CCE} ≈ _{CEC}.

Using a computer simulation, we calculated more precisely the thickness _{G} of the enzyme membrane at which _{CCE} = 1 for different enzymatic rates _{2}. Accepting _{1} = 1 μmol/(cm^{3}s) it was found that _{G} ≈ 0.0009 at _{2} = 100, _{G} ≈ 0.003 at _{2} = 10, and _{G} ≈ 0.009 cm at _{2} = 100 nmol/(cm^{3}s). These values of the membrane thickness compare favourably with values of the thickness _{max} at which the steady-state current as a function of the membrane thickness

Consequently, for a low substrate concentration the thickness _{G} of the enzyme membrane at which _{CEC} = 1 can be precisely enough expressed as _{G} ≈ 1.5 _{σ}, where _{σ} was defined in (_{1} and _{2}, if only the normalized substrate concentration _{0N} is less than unity.

For comparing the time of a steady-state amplified biosensor response with the steady-state time of the response without amplification, we introduce a prolongation (_{m}(_{1}, _{2}) is the steady-state time of the triggering biosensor acting in mode _{i} of the enzyme E_{i}, _{CE}(_{1}) is the steady-state time of the corresponding CE biosensor at the maximal enzymatic rate _{1}. Since the action of the CE biosensor can be simulated as an action of a CEC biosensor accepting _{2} = 0, we assume _{CE}(_{1}) = _{CEC}(_{1}, 0).

_{1} = 1 μmol/(cm^{3}s) and different values of _{2}. One can see in _{CEC} as well as _{CCE}) is a non-monotonous function of the thickness

In the cases of thin enzyme membranes (

In the case of the CEC mode, the slight influence of the maximal enzymatic rate _{2} on _{CEC} can be noticed in _{2} on _{CCE} is observed in the case of CCE action mode. Additional calculations showed that the response time prolongation slightly depends on the substrate concentration _{0} as well as the maximal activity _{1} of the enzyme E_{1}.

The mathematical model (

The steady-state current _{CEC} of a biosensor acting in the CEC mode and the steady state current _{CCE} of a biosensor acting in the CCE mode are monotonous by increasing functions of both maximal enzymatic rates: _{1} and _{2} of enzymes E_{1} and E_{2}, respectively (_{CEC} and _{CCE}, of trigger biosensors were determined mainly by the enzymatic rate _{2} (_{2} is a critical factor for the biosensor current in the case of CCE mode, _{CCE} → 0 as well as _{CCE} → 0 if _{2} → 0. In the case of a CEC biosensor, the decrease of activity _{2} causes the decrease in gain _{CEC}; however, _{CEC} stays greater than unity, _{CEC} → 1 if _{2}→ 0.

Both signal gains, _{CEC} and _{CCE}, are most significant when the normalized concentration _{0N} of the substrate is less than unity (_{1}) of enzyme E_{1} and high activity (rate _{2}) of enzyme E_{2}. In the cases of relatively high maximal enzymatic activity _{1}, the signal amplification is stable only for low concentrations of the substrate.

In both biosensors acting modes, an insignificant amplification of the signal is observed if the diffusion modulus ^{2}, calculated with the enzymatic rate _{2}, is less than unity, i.e. the kinetics of enzyme E_{2} controls the biosensor response. The gain _{CCE} becomes even significantly less than unity if ^{2} ≪ 1. For this type of biosensors, at a low substrate concentration, _{0N} < 1, the gain _{CCE} exceeds unity only when

In the cases where the significant amplification of the signal of a triggering biosensor is achieved, the response time is up to several times longer than the response time of the corresponding biosensor acting without triggering (

The models developed are permitted to build new trigger biosensors (in particular, by utilizing the CCE scheme). A highly sensitive hydrogen peroxide biosensor is under development and signal amplification has found the experimental confirmation.

This work was supported by Lithuanian State Science and Studies Foundation, project No. C-03048. The authors are grateful for the assistance of Dr. R. Lapinskas.

The profiles of the normalized concentrations of substrate (_{N}) and products (_{1N}, _{2N}) in the enzyme membrane of a CEC biosensor at the maximal enzymatic rate _{1} = _{2} = 100 nmol/(cm^{3}s), _{0N} = 0.2,

The profiles of the normalized concentrations in the enzyme membrane of a CCE biosensor at time

The steady-state current versus the maximal enzymatic rates _{1} and _{2} of the biosensor acting in CEC mode, _{0N} = 0.2,

The steady-state current versus _{1} and _{2} of the biosensor acting in CCE mode at the same conditions as in

The signal gain _{CEC} versus the maximal enzymatic rates _{1} and _{2} of the biosensor acting in the CEC mode at the conditions defined in

The signal gain _{CCE} versus the maximal enzymatic rates _{1} and _{2} of the biosensor acting in the CCE mode at the conditions defined in

The signal gains _{CEC} (1, 3, 5, 7) and _{CCE} (2, 4, 6, 8) vs. the substrate concentration _{0N} at the maximal enzymatic rates _{1}: 100 (1-4), 1 (5-8) and _{2}: 100 (1, 2, 5, 6), 10 (3, 4, 7, 8) nmol/(cm^{3}s),

The signal gains _{CEC} (1-3) and _{CCE} (4-6) versus the membrane thickness _{2}: 100 (1, 4), 10 (2, 5) and 1 (3, 6) nmol/(cm^{3}s); _{1} = 1 μmol/(cm^{3}s), _{0N} = 0.2.

The increase of response time _{CEC} (1-3) and _{CCE} (4-6) versus the membrane thickness