Chapter 4
Mathematical Modeling of Enzymatic Sensors
4.1 Introduction
A particular feature of enzymatic biosensors is the dynamic character of the sensor processes. Once the substrate-containing solution comes into contact with the sensor, conversion of substrate under the catalytic action of the enzyme begins and induces concentration gradients. As a result, diffusion of both reactant and product is sustained as long as the substrate is available. Consequently, modeling of enzymatic biosensors is a problem of mass transfer coupled with a chemical reaction that is catalyzed by an immobilized catalyst. From this standpoint, an enzymatic sensor can be viewed as a small enzyme reactor integrated with a transduction device.
The expected output of the modeling approach is the response function that gives the correlation between the sensor response and the substrate concentration in the test solution. Modeling reveals the essential working parameters and is of particular importance for the rational design of the sensor. Also, it allows predicting important features, such as the extent of the linear calibration range, the limit of detection, the response time and possible interferences.
The key assumption in the present approach is that the transduction response is a function of the concentration of a product at the transducer interface. Therefore, the main goal of the mathematical modeling is to find a relationship between the product concentration at the transducer surface and the substrate concentration in the test solution.
It will be assumed that the transduction process occurs without consumption of the product. This assumption is reasonably fulfilled if the transduction is performed by potentiometric, conductometric or optical methods. Amperometric enzyme sensor do not always fulfill this condition and modeling methods for such sensors are therefore left for a later chapter (Chapter 15).
Clearly, a stable, time-independent response, results only under steady-state conditions, in which the concentration of any reactant and product is time-independent at any point. The steady-state condition is therefore a main assumption in further derivations. Mathematically, it is expressed by equating the rates for the chemical reaction and mass-transfer process relative to both substrate and product. As is typical of a sequential process in the steady state, the overall reaction rate is determined by the slowest step, which can be either the enzymatic reaction or a diffusion step.
Modeling of immobilized enzyme reactors is of outstanding interest in biotechnology and is comprehensively dealt with in various texts (e.g., [1–3]). A good introduction to the modeling of enzymatic sensors is available in ref. [4].
Functioning of an enzymatic sensor is determined to a great extent by the diffusion processes. That is why this chapter is organized in accordance to the diffusion conditions. The first two sections deal with limiting cases in which diffusion is localised either out of the enzyme layer (external diffusion) or inside the enzyme layer (internal diffusion). The final section addresses the case in which diffusion in both the solution and the enzyme layer are relevant to the sensor functioning.
4.2 The Enzymatic Sensor under External Diffusion Conditions
4.2.1 The Physical Model
The physical model of an enzymatic sensor functioning under external diffusion conditions is shown in Figure 4.1. This sensor consists of a homogeneous enzyme layer intercalated between the transducer surface and a membrane permeable to the substrate and the product. It is assumed that the substrate and product concentrations are constant within both the enzyme layer and in the test solution. An even concentration distribution within the solution phase is secured by stirring. In order to obtain an even distribution of concentration within the enzyme layer, the enzyme loading should be high and the diffusion through the membrane should be sluggish. A more accurate definition of the external diffusion conditions is given in Section 4.4.
Figure 4.1 Concentration profiles and mass fluxes in an enzymatic sensor under external diffusion conditions. p and s denote the concentration of the substrate and product in the solution, respectively; the same symbols with the subscript e indicate substrate and product concentrations in the enzyme layer. 
and 
are diffusive fluxes of the substrate and product, respectively.
Under the above conditions, concentration gradients occur only within the membrane. It is assumed for simplicity that concentration profiles within the membrane are linear.
In the framework of this model, the transduction is performed by detecting the product of the enzymatic reaction and the transduction process does not involve product consumption. The response signal depends on the product concentration within the enzyme layer and the goal of this approach is to derive an equation relating this parameter to the substrate concentration in the sample solution. Initially, the product is not present in the solution and the amount of product accumulated next into the solution is very low. Hence, the product concentration in the solution negligible during the run. As the mass-transfer process is limited to the sensor region outside the enzyme layer, it is said that the sensor is operated under external mass-transfer conditions.
The physical model depicted in Figure 4.1 pertains to a membrane-covered enzyme layer. However, this model applies also to membraneless sensors. In this case, the diffusion gradient is localized in a solution film adjacent to the enzyme layer according to the Nernst diffusion model.
4.2.2 The Mathematical Model
The functioning of enzymatic sensors is based on a sequential two-step process involving diffusion and enzymatic conversion of the substrate. In the steady state, the velocity of the enzymatic reaction is equal to the diffusion rates of the substrate and the product. The reaction velocity will be expressed by the volume reaction rate, which is the product of the reaction rate within the enzyme layer (v′) and the volume of the enzyme layer (V):
(4.1) 
The volume reaction rate (in mol s−1) indicates the number of moles of substrate converted within the whole enzyme layer per time unit. Assuming that the enzyme reaction follows Michaelis–Menten kinetics, the volume reaction rate is:
(4.2) 
where 
is the enzyme concentration in the enzyme layer, 
is the rate constant and 
is the Michaelis–Menten constant. Each of the above constants refers to the immobilized enzyme and can be different from the relevant constant determined in the solution phase.
The diffusion rate is expressed by the diffusive flux (J) that indicates the number of moles crossing in unit time a unit area cross section perpendicular to the diffusion direction; the flux unit is mol m−2 s−1. In the framework of the Nernst layer model, the diffusive flux is proportional to the concentration difference between the solution and the enzyme layer, the proportionality constant being termed the mass-transfer coefficient. Therefore, the substrate and product fluxes depend on concentration as follows:
(4.3) 
where 
and 
are the mass-transfer coefficients of the substrate and the product, respectively and p = 0. The mass-transfer coefficient is the quotient of the diffusion coefficient and the thickness of the diffusion layer:
(4.4) 
where 
and 
are the diffusion coefficients of the substrate and product, respectively, in the membrane, and 
is the thickness of the membrane.
The steady-state condition is derived from the mass-conservation law and reads:
where A is the surface area of the enzyme layer. Upon dividing Equation (4.5) by 
one obtains:
The quantity 
(in mol m−2 s−1) is the surface-normalized reaction rate. It represents the amount of substrate converted per unit time and unit surface area. The conservation Equation (4.6) can now be formulated as follows:
Two limiting kinetic cases are possible depending on the 
ratio, that is, either 
or 
.
4.2.3 The Zero-Order Kinetics Case
In the zero-order kinetic case, 
and, therefore, the Michaelis–Menten constant in the kinetic equation can be neglected. Thereby, an equation formed by the last two terms in Equation (4.7) gives:
where 
is the maximum reaction rate for the immobilized enzyme. Equation (4.8) demonstrates that under zero-order kinetics conditions, 
(and, hence, the response signal) is independent of the substrate concentration in the solution but depends on the total concentration of the enzyme. Therefore, the determination of the substrate concentration cannot be accomplished with an enzymatic sensor functioning under zero-order kinetics conditions. However, the sensor response depends on the total concentration of active enzyme within the biocatalytic layer. Such conditions are favorable when using the enzymatic sensor to determine an enzyme inhibitor. Under the effect of the inhibitor, a fraction of the total enzyme amount loses its activity and the sensor signal decreases as a function of the inhibitor concentration.
In addition, this case is suitable when an enzyme is utilized as a label tag for an otherwise nondetectable compound. The recognition process brings the labeled analyte near the transducer and, in the presence of the substrate the sensor generates a signal depending on the concentration of the enzyme label. This signal is directly related to the analyte concentration.
The 
condition required in the above two applications can be fulfilled by proper adjustment of the substrate concentration that has to be added to the sample solution.
4.2.4 The First-Order Kinetics Case
In the first-order kinetics case, 
. In this instance, the limiting Michaelis–Menten equation for the first-order kinetics case will be used to define the surface-normalized reaction rate as follows:
(4.9) 
where
(4.10) 
where 
. At a constant enzyme concentration, 
represents the surface normalized pseudofirst-order rate constant. Further, by equating the diffusional flux of the substrate with the reaction rate, it results:
(4.11) 
This equation yields:
The above equation puts into evidence the following coefficient:
Using the 
coefficient, Equation (4.12) becomes:
The dimensionless coefficient 
(termed the substrate modulus for external diffusion) is a key parameter in the modeling of processes involving immobilized enzymes. By expanding the terms in Equation (4.13) one obtains an expression for 
that shows the effect of the kinetic, diffusional and geometric parameters:
According to its definition in Equation (4.13), the 
parameter depends on the rate constants of two consecutive processes: the transport of the substrate (
) and the enzymatic conversion of the substrate (
). Therefore, the physical meaning of the 
parameter can be rationalized as follows:
A value of 
greater than 1 indicates a fast chemical reaction preceded by an intrinsically slow diffusion step. At the 
limit, diffusion is so slow that the supply of substrate is much below the conversion potential of the enzyme layer and the overall reaction rate is wholly determined by the rate of substrate diffusion. Under these circumstances, any substrate molecule reaching the enzyme layer is instantaneously converted into the product and the sensor operates under diffusion control. In the opposite case (
) the supply capacity greatly exceeds the conversion capability of the enzyme layer. Therefore, the concentration gradient adjusts itself to a very low value in order to allow diffusion to keep pace with the sluggish chemical reaction. In this situation, the sensor process occurs under kinetic control.
It is now possible to derive an expression for the 
variable that determines the sensor response. To this end, the above expression for 
(4.14) will be substituted in an equation formed by the first two terms in Equation (4.7). Upon solving it for 
, one obtains:
Equation (4.17) proves that if 
, the product concentration in the enzyme layer is proportional to the substrate concentration in the solution, the proportionality constant being dependent on the 
coefficient. If 
, the proportionality constant in Equation (4.17) reduces to 
, which causes the kinetic parameters of the enzyme reaction to become irrelevant. Under such circumstances, the sensor functions under diffusion control. The 
variable assumes in this situation its maximum possible value for a given substrate concentration and imparts to the sensor the maximum sensitivity.
It is important to notice that under diffusion control, the response is independent of any parameter of the enzyme layer, which renders the sensor stable under long-term storage and operation and less sensitive to fluctuations arising from the manufacturing process.
If the condition 
is not fulfilled, the sensitivity is lower. Also, the response depends not only on diffusion parameters but also on the enzyme concentration as well as the thickness of the enzyme layer. In this case, the sensor response is sensitive to changes in pH or temperature (which affect the reaction rate) as well as to possible enzyme inactivation by degradation or inhibition.
In conclusion, in the first-order kinetics case, the product concentration in the enzyme layer is proportional to the substrate concentration in solution. If, in addition, the sensor operates under diffusion control (
), the product concentration in the enzyme layer is independent on the parameters of this layer, which renders the sensor reliable and resilient.
4.2.5 The Dynamic Range and the Limit of Detection under External Diffusion Conditions
It has been proved in the previous sections that substrate determination can be performed with the sensor functioning under the first-order kinetics regime that implies that the condition 
is fulfilled. As the 
value is out of direct control, it is important to define the limit of the linear range in terms of substrate concentration in the solution. To this end, the response equation should be derived with no restrictions in the value of the 
ratio. The starting point in this derivation is the mass-conservation equation formulated as follows:
where 
is the surface-normalized maximum reaction rate. Using dimensionless concentrations (
and 
) and solving for 
, one obtains from Equation (4.18) the following expression:
where 
is similar to the coefficient defined in Equation (4.15). In order to derive the concentration of the product in the enzyme layer, the following conservation equation with respect to dimensionless concentrations will be used:
(4.20) 
where 
is the dimensionless concentration of the product within the enzyme layer. By solving this equation, it follows that:
The above equation gives the product concentration in the enzyme layer as a function of the substrate concentration in the solution with no restrictions on the value of the 
ratio. This equation has been used to generate the curves in Figure 4.2 that indicate the 
relationship in the first-order kinetics regime. The curves in this figure demonstrate clearly that the extent of the linear response region increases with increasing 
coefficient. An estimation of the limit of the linear range can be obtained by setting 
and solving Equation (4.19) for S. One obtains thus the approximate limit of the linear range at 
. It is worth mentioning that the upper limit of the working range could be determined not only by the 
modulus but also by the transduction process itself.
Figure 4.2 Correlation between the product concentration within the enzyme layer (Pe) and the substrate concentration in solution (S) in the case of external diffusion control. Curves generated using Equation (4.21) with 
.
As far as the limit of detection is concerned, it is independent of 
but is determined by the intrinsic limit of detection of the transduction method.
Sensor sensitivity is determined by the pre-
factor in Equation (4.17). Since the sensor is as a rule designed such that 
, it follows that the sensitivity is practically independent of this factor. In agreement with this conclusion, the linear parts of the curves in Figure 4.2 do overlap.
Recalling the definition of 
, it is clear that this parameter can be tuned by adjusting the enzyme concentration in the biocatalytic layer, the thickness of the enzyme layer and the characteristics of the external membrane. The 
parameter increases with decreasing mass-transfer coefficient through the membrane, that is, with increasing membrane thickness. However, a decrease in the mass-transfer coefficient is accompanied by an increase in the response time.
In conclusion, a high value of the 
parameter is beneficial for the sensor response as well as for sensor resilience.
in Equation (4.16).
and 
in Equation (4.13) in order to derive an equation for the 
coefficient including all characteristic constants and geometrical parameters. Discuss the effect of each term in this equation on the 
parameter and comment on possible methods for increasing the 
value.
, and (b) 
and comment on the effect of the substrate concentration in the solution. For case (b), define the condition needed in order for the reaction rate to be proportional to the substrate concentration in the solution.
–S curve for α = 1, 10, and 50 and comment on the degree of utilization of the enzyme under first-order kinetics and external diffusion control.
), derive an equation relating the response to the substrate concentration in the solution. Comment on the effect of the 
coefficient on the sensor sensitivity.
). Comment on the effect of the 
coefficient on the sensor response and point out the advantage of a very high value of 
.4.3 The Enzymatic Sensor under Internal Diffusion Control
4.3.1 The Steady-State Response
In many instances, concentration gradients form inside the enzyme layer and the mass transfer within this region cannot be neglected. Such a situation arises if the enzyme layer is relatively thick or the diffusion coefficients within this layer are very small. The concentration profiles of the substrate and product in this case are shown in Figure 4.3. In order to simplify the mathematical treatment, it will be assumed that the concentration gradients out of the enzyme layer are negligible. This condition is fulfilled when the solution is well stirred and the membrane permeability is very high. Actually, this model fits best membraneless sensors.
Figure 4.3 Concentration (c) profiles for substrate and product in the case of an internal diffusion enzymatic sensor. Concentration gradients out of the enzyme layer are negligible. Substrate and product partition at interfaces is neglected. s and p represent substrate and product concentrations in the solution, respectively; 
and 
represent similar concentration in the enzyme layer.
The mathematical model will be obtained upon combining Fick's second law of diffusion with the expression of the reaction rate. Taking into account the conclusions of the previous section, it is convenient to consider the limiting case of first-order kinetics (
). Under these conditions, the differential equations for diffusion combined with chemical reaction assume the following forms:
and 
denote the diffusion coefficients within the enzyme layer for the substrate and product, respectively, and 
is the distance from the surface of the transducer. In the above equations, the left-hand term represents the local time variation of the concentration. This is caused by two processes: diffusion and the enzymatic reaction, whose contributions are expressed by the first and second terms on the right-hand side of the equation, respectively. The required mathematical solution of the above equations is the product concentration at the transducer surface (
) that determines the transducer response.
The limiting conditions are, as follows: for 
, 
and 
(that is, product concentration in the test solution is negligible); for 
and 
, 
(that is, neither the substrate nor the product is initially present in the enzyme layer). Assuming that the transduction occurs without material consumption, concentration gradients at 
are equal to zero.
In the steady state, concentrations are constant at any point within the enzyme layer. Therefore, the steady-state solution is obtained by nullifying the left-hand term in Equations (4.23) and (4.24). The solution of this problem demonstrates that the sensor response depends essentially on the dimensionless parameter 
defined in Equation (4.25) and identified as the Thiele modulus or the substrate modulus for internal diffusion.
Therefore, the Thiele modulus groups all relevant parameters of the enzyme layer. Often, the square of 
is utilized instead of 
. 
is called the enzyme loading factor (
) whereas chemical engineering identifies 
as the Damköhler number, Da:
(4.26) 
The physical meaning of 
can be inferred if the Equation (4.25) is rearranged such as to give the following relationship:
where 
is the surface-normalized rate constant and 
is the mass-transfer coefficient. According to Equation (4.27), 
is the quotient of two rate constants, namely the rate constant of the enzymatic reaction and the mass-transfer coefficient. Therefore, Equation (4.27) can be expressed as follows:
(4.28) 
It is clear that a high value of 
implies slow mass transfer, that is, the mass-transfer step determines the overall rate of the process. In the case of a low 
value, the overall reaction rate depends on kinetic parameters included in the 
constant and is independent of the mass-transfer parameters. Therefore, the 
parameter can be viewed as the degree of catalyst utilization.
Using the Thiele modulus, the differential equations for the steady-state conditions led to the following expression for the product concentration profile [4]:
This equation was used to plot the response factor (
) as a function of the dimensionless distance (
) for various values of the Thiele modulus, as shown in Figure 4.4A. This figure demonstrates that at low 
values the degree of conversion is rather low all over the enzyme layer, as a result of the sluggishness of the reaction. On the contrary, high 
values bring about a high conversion of substrate to product. At 
, as a consequence of the rapid chemical reaction, most of the substrate is converted within the right-hand portion of the enzyme layer, that is, before it moves far into the membrane enzyme film.
Figure 4.4 The enzymatic sensor under internal diffusion conditions and first-order kinetics. (A) Profile of the product concentration within the enzyme layer for various 
values. (B) Simulated response of the enzyme sensor under internal diffusion control. (
) for both graphs.
It is possible now to account for the sensor response. As before, it will be assumed that the response is a function of the product concentration at the transducer surface, that is, at 
. For this particular condition, Equation (4.29) gives:
This equation proves that the product concentration at the transducer surface (
) is proportional to the substrate concentration in solution. The proportionality constant depends on the ratio of diffusion coefficients. What is even more important is that the proportionality constant is also a function of the Thiele modulus. However, if 
, 
becomes exceedingly greater than unity and the proportionality constant in Equation (4.30) reduces to the ratio of the diffusion coefficient. From the physical standpoint, this limit corresponds to a slow mass transfer, which makes the overall process occur under internal diffusion control.
Assuming that the response is proportional to 
, the profile of the sensor response will be the same as the straight lines in Figure 4.4B. This figure demonstrates an increase in sensitivity with increasing the 
value. Nevertheless, the sensitivity reaches a limit at 
and any further increase of this parameter has no effect on the sensitivity. Besides, as 
and 
could be very different in some instances, the effect of the 
ratio on sensitivity should not be disregarded.
It should also be borne in mind that Equation (4.30) was derived under the assumption that 
. If the substrate concentration in solution is high enough to render this condition no longer applicable, the sensor response will deviate from the linear trend displayed in Figure 4.4B.
As in the case of external diffusion control, no restriction on the limit of detection arises as far as the reaction kinetics are concerned. Consequently, the limit of detection depends mostly on the response characteristics of the transduction process. This latter can also impose the upper limit of the linear range, even if the above-mentioned kinetic features allow, in principle, the attainment of a higher limit.
In conclusion, the sensitivity of an enzymatic sensor functioning under internal diffusion conditions increases with increasing Thiele modulus 
. According to Equation (4.25), the value of the Thiele modulus depends on the enzyme concentration within the enzyme layer, the thickness of this layer and the diffusion coefficient of the substrate. In turn, the diffusion coefficient is determined by the method of enzyme immobilization. It should be borne in mind that an increase of the thickness and a decrease of the diffusion coefficient result also in an increase in the response time.
A high value of the Thiele modulus brings also advantages as far as the long-term sensor response, stability and reproducibility are concerned. Equation (4.25) demonstrates that for 
, this modulus drops out from the response function and the response becomes independent of the parameters of the enzyme layer. Therefore, to a certain degree, alteration of the enzyme activity (by drgradation of inhibition) has no effect on the sensor response.
4.3.2 The Transient Regime and the Response Time under Internal Diffusion Conditions
The transient behavior of the sensor is noticeable immediately after initiating the sensing process. In order to investigate transient behavior, the substrate should be initially absent in both the solution and the enzyme layer. In order to trigger the sensing process, the substrate is added to the well-stirred solution at 
. It is clear that the response signal will vary with time before reaching a constant value, which is typical of the steady state.
The next treatment is adapted from ref. [5] in which the modeling of the transient behavior has been accomplished at the first-order kinetics limit (
) under the assumption that substrate and product have identical diffusion coefficients within the enzyme layer (
). By solving the time-dependent Equations (4.23) and (4.24), the product concentration in the enzyme layer has been obtained as a function of time and distance from the transducer surface (Equation (4.12) in [5]), which can be put in the following form:
The first term in this equation represents the steady-state response (see Equation (4.29). The second term is a function of the dimensionless time variable 
, where t is the time and 
is the diffusional time constant, which is defined as:
(4.32) 
The time-dependent term in Equation (4.31) also includes the Thiele modulus. This term decreases asymptotically with time and becomes close to zero when the steady state is approached.
Equation (4.31) has been used to simulate the profile of the product concentration within the enzyme layer at various stages after initiating the enzymatic reaction. The results of the simulation are shown in Figure 4.5. The curve for 
represents the profile of the product concentration at an early stage after initiating the sensing process. Under these conditions, substrate molecules entering the enzyme layer are immediately converted and a product concentration gradient develops. Consequently, the product undergoes diffusion towards the interior of the enzyme layer. However, due to the inherent slowness of diffusion, the product concentration fades away at some distance from the interface. As the process proceeds further, the gradient of the product concentration increases. Thereby the diffusion rate is enhanced, prompting the product to spread further within the enzyme layer (see the intermediate curves in Figure 4.5A). When the time elapsed is such that 
, the second term in Equation (4.31) becomes negligible and the steady-state profile is set up. This situation is represented by the uppermost curve in Figure 4.5A, which overlaps the points plotted by means of the steady-state equation (4.29).
Figure 4.5 The enzymatic sensor in the transient regime. (A) Product concentration profiles within the enzyme layer at various stages for 
. Circles indicate the steady-state profile according to Equation (4.29). (B) Transient response of an enzymatic sensor under internal diffusion control for 
and various values of the diffusional time constant.
As the sensor response depends on the product concentration at the transducer surface, it is of interest to inspect the time variation of this variable. Equation (4.31) can be used to this end after setting 
. The simulated time variation of the 
ratio is displayed in Figure 4.5 B for 
and several values of the diffusion time constant. At the selected value of 
, the overall reaction rate is determined by the diffusion step. The curves in Figure 4.5 B demonstrate that the greater the value of the time constant, the longer is the time needed for the steady state to set up.
The response time of the sensor can be estimated by putting Equation (4.31) in an approximate form, which is valid only when the steady state is approached (that is, for 
). According to ref. [5], the time dependence under these conditions is represented by the following equation:
This equation proves that shortly before the steady state is approached, the time variation of the response is independent of the enzyme loading. It depends only on the diffusional parameters included in the 
constant. An expression for the response time (
) can be derived by assuming that the response time is the time elapsed until the second term in Equation (4.33) drops to a negligible value such as 0.01, that is about 1% of the final 
value. Based on this assumption, the response time is obtained as:
(4.34) 
Accordingly, the steady state is practically set up when the time elapsed after triggering the sensor process reaches twice the value of the diffusional time constant. This conclusion is in good agreement with the plots in Figures 4.5A and B. Therefore, the response time can be modified by changing the diffusion-related parameters. Among them, the thickness of the enzyme layer is the most susceptible to adjustment. The diffusion coefficient in turn depends essentially on the enzyme immobilization method. It could be in the region of 
cm2 s−1 if an enzyme solution is entrapped between the transducer surface and a semipermeable membrane, but it is much lower if the enzyme is embedded in a gel or is incorporated in the pores of a solid material.
Summing up, under internal diffusion control, the transient response depends on both the enzyme loading factor and the diffusional time constant. Nevertheless, the effect of enzyme loading is manifest only at the beginning of the transient regime and the actual response time depends only on diffusion-related parameters, namely the thickness of the enzyme layer and the diffusion coefficient of the substrate within the enzyme layer.
It should also be kept in mind that the above results were derived under simplifying conditions. An actual sensor may deviate more or less from this model. Nevertheless despite its approximate character, this approach indicates the main factors affecting the response time and also the expected effects of these factors.
) and the maximum attainable mass transfer rate (which occurs when 
).
= 1, 25, and 100, calculate the per cent variation of the sensor response due to the change in enzyme activity for each of the above values of 
. Assume that 
.4.4 The General Case
4.4.1 The Model
This section addresses the general problem of a catalytic reaction in an immobilized enzyme layer coupled with diffusion in both the enzyme layer and an adjacent layer, which could be a semipermeable membrane or a solution layer within which significant concentration gradients develop. The following approach is based on the theoretical treatment in ref. [6]. The physical model of the considered system is presented schematically in Figure 4.6.
Figure 4.6 Concentration (c) profiles for a biocatalytic layer attached to the surface of a transducer. s and p denote concentrations of the substrate and product, respectively. Subscripts m and e to p and s indicate concentration in the membrane and the enzyme layer, respectively.
This model takes into account the effect of partition at the interface between the catalytic and noncatalytic layers. Partition causes a sudden change in the concentration at this interface and can be quantified by the partition constants of the substrate (
) and the product (
) that are defined as follows:
(4.35) 
The meaning of symbols in these equations is given in Figure 4.6.
It is also assumed that the reaction obeys the first-order kinetics (
) and occurs according to a Michaelis–Menten-type mechanism in which each substrate molecule gives rise to 
product molecules.
(4.36) 
No provision is made for a possible pH change in the enzyme layer. Hence, it is assumed that the pH is kept constant by means of a pH buffer system.
Diffusion through the enzyme layer is described by modified Fick's Equations (4.23) and (4.24). At the enzyme layer/membrane interface, concentration profiles show discontinuities due to partition, which should be accounted for in the relevant limiting conditions of Fick's equations. The transport in the membrane layer is modeled by means of the Nernst layer concept that gives the following expressions for the diffusional fluxes:
(4.37) 
(4.38) 
where 
and 
are the mass-transfer coefficients of the substrate and product, respectively, in the membrane.
Assuming that the product concentration in the solution is negligible, the above model leads to an equation relating the product concentration at the transducer surface (
) to the substrate concentration as in the solution follows:
The Thiele modulus is immediately recognizable in the above equation that includes several additional dimensionless parameters denoted by 
, 
, and 
.
is the Biot number that quantifies the relative preponderance of internal or external diffusion:
where 
.
Large 
values imply 
and indicates that internal diffusion is very slow compared with the external diffusion. This is typical of an enzymatic sensor operating under internal diffusion control. The case of very low 
values implies much slower diffusion within the membrane compared with the diffusion within the enzyme layer, which leads to the external diffusion control case. The Biot number allows therefore making of a clear-cut distinction between internal and external diffusion regimes as limiting cases. Using the definitions of 
, 
and 
coefficients, it is easy to prove that these coefficients are interconnected by means of the Biot number as follows:
(4.41) 
Each of the two 
coefficients in Equation (4.40) indicates the rapidity of the substrate diffusion relative to that of the product and will be termed lag factors. The internal lag factor is defined as:
(4.42) 
A large value of 
results when the diffusion coefficient of the product in the enzyme layer is much lower that that of the substrate. Under these circumstances, product depletion by diffusion out of the enzyme layer is a slow process and the product can accumulate within the enzyme layer. This leads to an increase in sensitivity.
The external lag factor (
) is identified as:
(4.43) 
A large value of 
indicates a slow diffusion of the product across the membrane and leads consequently to an enhanced product concentration within the enzyme layer.
4.4.2 Effect of the Biot Number
Equation (4.39) will be used next to examine the effect of the Biot number and understand in which way the limiting situations of internal or external diffusion can be attained. To this end, it is convenient to made to following simplifying assumptions: 
, 
, 
. Thus, the effect of partition and unevenness of diffusion coefficients is removed and Equation (4.39) simplifies to:
For the purpose of this discussion it is useful to mention that for 
we have 
and for 
, 
.
As already mentioned, the internal diffusion regime is attained at very high 
values, more precisely, if 
, which implies that 
. Therefore, under the internal diffusion regime, the response factor Equation (4.44) becomes:
Taking into account the above simplifying assumptions, this equation is similar to Equation (4.30) derived under the hypothesis of a purely internal diffusion regime.
The external diffusion regime occurs if 
is sufficiently low to satisfy the condition 
. Taking into account this condition, a limiting form of Equation (4.44) can be derived. However, this limiting equation is somewhat ambiguous because it can yield negative values for the response factor. That is why Equation (4.17) which was derived under the assumption of purely external diffusion control will be further compared with the more general Equation (4.44) upon assuming that 
. To this end, a change of variable in Equation (4.17) will be operated in order to substitute the parameter 
by its expression in Equation (4.40). As 
and the concentration is evenly distributed across the enzyme layer (that is, 
), one obtains the following expression for the response factor under external diffusion conditions:
In order to assess the interplay between internal and external diffusion, Equation (4.44) was used to plot the response factor vs. 
for selected values of the Biot number (solid lines in Figure 4.7). A typical curve for intermingled diffusion is that for 
(curve 3). An increase in the Biot number results in a shift of the curve toward higher values of 
until the limit of the internal diffusion regime is reached for 
(curve 4). For comparison, data obtained with the limiting Equation (4.45) are plotted as dots in Figure 4.7. It is clear that the general Equation (4.44) and the limiting Equation (4.45) lead to concordant results.
Figure 4.7 Effect of the Thiele modulus 
on the product concentration at the transducer surface for various values of the Biot number 
. Solid lines were generated with Equation (4.44) for combined internal and external diffusion. Dots represent the limiting case of internal diffusion (Equation (4.45)). Circles were generated with the approximate Equation (4.46) for external diffusion. Biot number: (1) 0.01; (2) 0.1; (3) 1; (4) 25.
The external diffusion case occurs at subunity values of 
and is represented by curve 1 in Figure 4.7, which is plotted for 
. This curve overlaps the circles plotted with the limiting Equation (4.46). For 
(curve 2) small differences between the two series of data can be noticed.
The above discussion demonstrates clearly that it is the Biot number 
that dictates the prevalence of internal or external diffusion. Internal diffusion control is dominating at 
, whereas external diffusion control operates at 
. The above figures can change if the effect of partition and lag coefficients are also taken into account, but the trends illustrated in Figure 4.7 are applicable even in such situations.
Keeping in mind that the maximum sensitivity is achieved when 
, Figure 4.7 demonstrates that for this limit to be achieved, a higher enzyme loading is needed with an increase in 
. While the internal diffusion regime requires that 
in order to attain the maximum sensitivity, this limit can be achieved at lower 
values in the case of external diffusion. This feature can be rationalized if we take into account the fact that the external membrane brings a limitation to the access of the substrate to the catalytic layer. Therefore, a lower catalytic potential is in this case required in order to reach full conversion of the substrate to product. Clearly, a high diffusion resistance in the external membrane does not affect the sensitivity if 
, but it is beyond any doubt that the lower the value of 
the longer the response time. Figure 4.7 proves, in addition, that the limiting Equation (4.17) is in good agreement with the general Equation (4.44) if 
.
The above discussion has pointed out the importance of the Biot number in deciding the localization of concentration gradients either within the enzyme layer or within an external layer. In order to distinguish the actual parameters that determine the value of the Biot number, Equation (4.40) will be reformulated as follows:
(4.47) 
It is clear that the simplest way to alter the value of the Biot number relies on adjusting the layer thicknesses. Thus, a very thin enzyme layer favors the external diffusion regime, whereas internal diffusion is expected to dominate when the thickness of the external membrane is very low or substrate diffusivity within it is very high. If the enzyme layer is in direct contact with the solution, the thickness of the external diffusion layer is determined by the hydrodynamic conditions of the solution and vigorous stirring results in a reduced thickness.
4.4.3 Effect of Partition Constants and Diffusion Coefficients
The following discussion is devoted to an analysis of the effect of partition constants and diffusion parameters that were neglected in the previous section. This analysis can be conveniently performed if it is assumed that 
because under such conditions the response factor approaches a value close to one (Figure 4.7). Since, in this instance, 
and 
, Equation (4.39) assumes the following form:
According to this equation, diffusional and partition parameters intermingle in such an intricate way that it is impossible to distinguish the effect of each of them. That is why only the extreme cases of internal or external diffusion will be considered next. The external diffusion regime is achieved if 
is so small that 
. If the condition 
is also fulfilled, Equation (4.48) becomes:
(4.49) 
For the internal diffusion regime to occur, 
must be very high. In this case, 
. Assuming that 
, the following equation results from (4.48):
(4.50) 
The above equations prove that, in the case of the external diffusion regime, the response factor is proportional to the partition constant of the product, whereas the substrate partition affects the response only in the case of the internal diffusion regime. It is proved also that the response is proportional to the lag factor within the layer that determines the diffusion rate.
4.4.4 Experimental Tests for the Kinetic Regime of an Enzymatic Sensor
It was demonstrated above that the response of an enzymatic sensor is determined by the interplay of kinetic and diffusional factors. From the practical standpoint it is important to distinguish what kind of process represents the rate-determining step in the sensor process.
Kinetic control of the overall process is indicated by a strong increase of the sensor response with increasing temperature, provided that the transduction process is less affected by this parameter. This interpretation is based on the fact that the rate of a chemical reaction is strongly dependent on temperature (according to the Arrhenius equation) whereas the temperature has a much smaller effect on diffusion coefficients.
External diffusion control can be verified by checking the effect of the thickness of the external membrane. In the absence of the membrane, external diffusion is localized within a liquid film (Nernst layer) in which strong concentration gradients develop. If external diffusion control is present, the thickness of the Nernst layer can be altered by changing the hydrodynamic conditions such as the stirring rate in batch analysis or the fluid flow rate in flow analysis systems.
4.5 Outlook
The overall process occurring in an enzymatic sensor involves first the catalytic conversion of the substrate to products. As the enzyme is present in an immobilized form, diffusion of the substrate and product are coupled with the enzymatic reaction.
The sensor response depends to a large extent on the kinetics of the enzymatic reaction. Suitable conditions for substrate determination are provided by first-order kinetics. In this case, the sensor response is proportional to the substrate concentration in the sample.
The substrate diffuses from the solution to the enzyme layer, whilst the products travel from the enzyme layer to the solution. The enzyme layer may be coated with a semipermeable membrane or may be in direct contact with the test solution. In both cases, concentration gradients form in the film adjacent to the enzyme layer. Diffusion processes occur therefore in three distinct regions: the enzyme layer, the adjacent film and the bulk of the solution. The diffusion rate in each layer is characterized by the mass-transfer coefficient, which indicates the transport capacity. The overall diffusion rate is determined by the region of lowest transport capacity. Owing to the sluggishness of diffusion in such a region, high concentration gradients develop within it while in adjacent regions of high transport capacity, concentration gradients could be negligible. Therefore, two limiting cases can be distinguished: external diffusion and internal diffusion control. In the first case, the low transport capacity in the external (membrane) layer determines the overall mass-transfer rate, whilst in the second case it is the transport capacity in the enzyme layer that controls the overall mass-transfer process.
The interplay of internal and external diffusion is determined by the dimensionless Biot number. A high value of the Biot number implies sluggish internal diffusion that determines internal diffusion control. Conversely, low Biot number values correspond to sluggish external diffusion and lead to external diffusion control.
In general, the response of the enzymatic sensor depends on the enzyme concentration and the rapidity of the enzymatic reaction (through the 
rate constant). It depends also on the diffusion rate within the layer with the lowest transport capacity. Finally, the response depends on the thickness of the enzyme layer that determines the total amount of enzyme per surface area unit. The effect of all these factors is synthetically expressed by particular dimensionless parameters, namely the substrate modulus in the case of external diffusion control and the Thiele modulus in the case of internal diffusion control.
In the general case, which corresponds to near-unity values of the Biot number, diffusion in both the enzyme layer and in the membrane should be considered. As the substrate modulus and the Thiele modulus are interdependent, the response is determined in general by the Biot number and one of the above moduli.
The above treatment refers to the case in which the enzyme kinetics obeys the Michaelis–Menten mechanism and the sensor response is determined by the product concentration at the transducer surface. Despite its limitations, an examination of this particular model system allowed the functioning principles of an enzymatic sensor and the parameters that determine the sensor response to be determined. A comprehensive survey of enzyme sensor modeling is available in ref. [7].
1. Engasser, J.M. and Horvath, C. (1976) Diffusion and kinetics with immobilized enzymes, in Immobilized Enzyme Principles (eds E. Katchalski-Katzir, L.B. Wingard, and L. Goldstein), Academic Press, New York, pp. 127–220.
2. Goldstein, L. (1976) Kinetic behavior of immobilized enzyme systems, in Immobilized Enzymes (ed K. Mosbach), Academic Press, New York, pp. 397–443.
3. Illanes, A., Fernandez-Lafuente, R., Guisan, J.M. et al. (2008) Heterogeneous enzyme kinetics, in Enzyme Biocatalysis: Principles and Applications, Springer, Dordrecht, pp. 155–203.
4. Carr, P.W. and Bowers, L.D. (1980) Immobilized Enzymes in Analytical and Clinical Chemistry: Fundamentals and Applications, John Wiley & Sons, New York.
5. Carr, P.W. (1977) Fourier-analysis of transient-response of potentiometric enzyme electrodes. Anal. Chem., 49, 799–802.
6. Blaedel, W.J., Boguslaski, R.C., and Kissel, T.R. (1972) Kinetic behavior of enzymes immobilized in artificial membranes. Anal. Chem., 44, 2030–2037.
7.Baronas, R., Kulys, J., and Ivanauskas, F. (2009) Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians. Springer Netherlands, Dordrecht.