# Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro-and nanoelectrochemical sensors

Reina Dannaoui, Xiao-Ke Yang, Wei-Hua Huang, Irina Svir, Christian Amatore, Alexander Oleinick

► To cite this version:

Reina Dannaoui, Xiao-Ke Yang, Wei-Hua Huang, Irina Svir, Christian Amatore, et al.. Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro-and nanoelectrochemical sensors. Electrochimica Acta, 2023, 473, pp.143425. 10.26434/chemrxiv-2023-3n4rp. hal-04272699

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Importance of diffusional constraints for the quantitative evaluation of
calibration curves of enzymatic micro- and nanoelectrochemical sensors<sup>‡</sup>

Reina Dannaoui,¹ Xiao-Ke Yang,² Wei-Hua Huang,² Irina Svir,¹,³ Christian Amatore,¹,⁴,*
Alexander Oleinick.¹,*

¹ PASTEUR, Département de Chimie, Ecole normale supérieure, PSL Université, Sorbonne Université, CNRS, 24 rue Lhomond, Paris 75005, France.

² College of Chemistry and Molecular Sciences, Wuhan University, Wuhan 430072, China.

³ Design Automation Department, Kharkiv National University of Radio Electronics, 14 Nauky Avenue, Kharkiv 61166, Ukraine.

⁴ State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China.

**Keywords:** Enzymatic nanoelectrode, calibration curve, Lineweaver-Burk plots, kinetic modelling, Michaelis-Menten related mechanisms.

**Abstract**

Electrochemical calibration curves recorded at enzyme-modified micro- or nanoelectrodes are often quantitatively analysed using graphical approaches. In this case the information is directly derived from the calibration curve using a frequent practice although it is rooted on extrapolating the procedures valid when the enzymes as well as their substrate and cofactors are homogeneously distributed in a solution. Here, using a simple but highly

<footnote>‡ Dedicated to Professor and Academician Hong-Yuan Chen to celebrate his career and honor his important contributions in the field of biosensors.</footnote>
representative model we demonstrate that this practice yields incorrect interpretation of
the experimental results even for simple Michaelis-Menten mechanisms. This model makes
it possible to establish correct relationships linking calibration currents and bulk substrate
concentrations by a simple method allowing to account for the biases due to diffusional
constraints.

# 1. Introduction

Enzyme modified electrodes allow to extend the limits of application of conventional electrochemical analytical sensors by making possible the detection of many non-electrochemically active compounds and/or by increasing the selectivity and sensitivity of the electrochemical detection [1-4]. This permits the selective detection and monitoring of biologically active molecules in particular under complex in vivo conditions possibly involving several electroactive interferents.

The performance and limitations of these sensors are often pre-evaluated or post-evaluated
by recording calibration curves that relate the current intensity of the sensor to the
concentration of the target analyte in a series of bulk test solutions. This is perfectly fine
whenever these sensors are intended to be used to study the compositions of other bulk
solutions whose concentrations may vary slowly provided they remain homogeneous
around the sensor. Indeed, when used under such conditions, the recorded calibration
curves feature intrinsic characteristics of the sensors, viz. of the enzymatic conversion and
the transport supply of the substrate to the sensor surface. Nowadays, highly miniaturized
sensors are more often designed to detect and quantify transient releases of analytes under *in vivo* or *ex vivo* conditions [5-7]. Under these conditions the problem will be even more severe due to complex and intermittent diffusion patterns encountered in such systems.

In this work, we first wish to highlight the practical importance of these questions by considering a canonical Michaelis-Menten mechanism (Figure 1) to illustrate the drastic role of diffusional constraints on the responses of enzymatic sensors under stationary or quasi-stationary conditions and their analyses. Our analysis aims to analytically evaluate the consequences of these constraints. Note that similar attempts have already been published, although with different focus and format (see e.g. refs. [4, 8, 9]). The central question of the problem at hand is introduced in section 2 “Preliminary considerations…” and is treated quantitatively in the case of a simple Michaelis-Menten mechanism. However, strictly sensu, this is rarely applicable to describe the performance of electrochemical sensors which generally involve a cosubstrate. For this reason, our approach is then extended to treat ping pong bi-substrate enzymatic mechanisms with or without inhibition.

## 2. Preliminary considerations stressing the role of diffusional constraints.

Let us consider here that the system at hand involves a simple canonical Michaelis-Menten (MM) enzymatic mechanism (Scheme 1) converting a non-electroactive substrate *S*, present at concentration [*S*]$_{bulk}$ in the analyzed bulk solution, into an electroactive product *P* detected at a micro- or nanoelectrode surface on which a thin enzyme coating is deposited (Figure 1) [7]. Note that by "thin" we mean that the thickness of the enzyme layer is extremely small relatively to the dimension of the electrode, therefore also vs. that of the diffusion layers which spontaneously establish in a few seconds at most in the enzyme-free
solution surrounding the electrode [10-12]. Therefore, one may rightly consider that most of the current flow stems from the electrolysis of the product generated in the enzyme layer without significant losses due to diffusion towards the bulk solution [8].

On the other hand, owing to the flux conservation of the substrate one has at any time:

$$J_S^{sol} = rate_{enz} \qquad (1)$$

where $J_S^{sol}$ is the diffusion flux of the substrate from the solution to the thin enzymatic layer where it is consumed with an overall rate of enzymatic conversion per unit of electrode surface, denoted $rate_{enz}$. Denoting $[E]_0$ the immobilized enzyme concentration, $K_M = (k_{cat} + k_b)/k_f$ the usual “Michaelis constant” and $[S]_{enz}$ the mean substrate concentration within the enzyme layer (Scheme 1), one has:

$$rate_{enz} = \frac{k_{cat}[E]_0}{1+K_M/[S]_{enz}} \qquad (2)$$

for a simple canonical Michaelis-Menten enzymatic mechanism without product inhibition [13, 14]. Note that $k_{cat}[E]_0$ represents the maximum conversion rate per unit of surface area, noted $V_{max}^{enz}$, achievable by the immobilized enzyme when $[S]_{enz} \gg K_M$.

<caption>[Fig.1]</caption>

$$E + S \underset{k_b}{\overset{k_f}{\rightleftharpoons}} ES \quad \underset{k_{cat}}{\overset{k_{cat}}{\rightleftharpoons}} E + P$$

$$P \underset{\text{current}}{\overset{ne^-}{\rightleftharpoons}} Y$$
**Scheme 1.** Canonical Michaelis-Menten (MM) mechanism with product electrochemical detection.

It is important to note that Eq. (2) involves the mean concentration $[S]_{enz}$ of the species S in the enzymatic layer and not that, $[S]_{bulk}$, in the bulk solution [4]. Actually, $[S]_{enz}$ depends on $[S]_{bulk}$ and the current flow, indeed:

$$i = nFA \times f([S]_{bulk} - [S]_{enz}) \quad (3)$$

where f is a function that depends on the diffusion profile of S inside the diffusion layer, A is the global surface area of the enzyme layer, n the number of electrons transferred per product molecule, F is the Faraday constant. It follows that:

$$[S]_{enz} = [S]_{bulk} - g \left( \frac{i}{nFA} \right) \quad (4)$$

where g is the inverse function of f. Therefore, whenever one wants to relate the current i to $[S]_{bulk}$, as is performed in the usual practice, the following equation must be used:

$$i = nFA \cdot rate_{enz} = nFA \frac{V_{max}^{enz}}{1 + K_M / \{[S]_{bulk} - g\left(\frac{i}{nFA}\right)\}} \quad (5)$$

instead of the oftenly used improper formulation [5-7, 15-18]:

$$i = nFA \frac{V_{max}^{enz}}{1 + K_M / [S]_{bulk}} \quad (6)$$

adopted within the context of analyses through Lineweaver-Burk plots, Eq.(7):

$$\frac{nFA}{i} = \frac{1}{V_{max}^{enz}} + \frac{K_M}{V_{max}^{enz}} \times \frac{1}{[S]_{bulk}} \quad (7)$$

However, Eq.(5) establishes that the real relationship between current and $[S]_{bulk}$ is:
$$ \frac{nFA}{i} = \frac{1}{V_{max}^{enz}} + \frac{K_M}{V_{max}^{enz}} \times \frac{1}{[S]_{bulk} - g(\frac{i}{nFA})} \quad (8) $$

which clearly confirms that contrary to what is considered in the usual practice, 1/i cannot vary linearly with 1/[S]bulk even for an electrochemical enzymatic sensor relying on a canonical Michaelis-Menten mechanism, i.e., onto the simplest rate law given in Eqn. (2). Equations (7,8) highlight the critical role of the g(i/nFA) function whose exact structure depends on the system under the scrutiny. In fact, non-linear Lineweaver-Burk experimental plots as well as Hanes-Woolf ones were also reported for enzymatic/heterogeneous biosensors (see e.g. refs.[19, 20]) being presumably due to the effect of g(i/nFA) function.

Therefore, it is the purpose of this work to precisely investigate this issue in detail in the following. This will be performed for the above MM canonical mechanism (Scheme 1) as well as for a series of other important enzymatic mechanisms of general interest for their involvement in bioelectroanalytical sensors. Note also that although they will be derived for a cylindrical micro- or nanosensor, the following results are readily transposable to other types of electrodes leading to steady state or quasi-steady state diffusional regime using previously established equivalence relationships [10, 12, 21]. As was noted at the end of the Introduction the above equations are obtained in the same spirit as was reported previously by, e.g., Bartlett et al [4], Lyons [9] and Kulys [8]. However, the results obtained below, are providing different insights so this study does not overlap with the previous ones and are rather to be seen as complementary.

### 3. Results and Discussion

To simplify the formulation of the models and of their ensuing key equations and focus on
the real issues at hand, the two following assumptions are made: (i) the partition coefficients between the enzymatic layer and the solution are equal to unity for all considered species; (ii) diffusion coefficients of substrate and products in the solution are set equal to their average value, *D*. It should be noted that using the proper values for partition coefficient and/or diffusivities may change the overall systems responses by introducing adequate multiplicative factors applying to [$S$]$_{enz}$, viz., can be viewed as being integrated in $K_M$ values. Yet, the general validity of the approach developed hereafter will not be affected since anyway $K_M$ value is not supposed to be known.

Finally, it worth to note that the flux balance analysis given below represents a usual practice for the analysis of the enzymatic systems (see. e.g. refs.[4, 8]), however we provide all the equations for clarity and consistency with our final results vs. the approach we develop in this work.

## 3.1 Canonical Michaelis-Menten mechanism

Figure 2a displays schematically the concentration profiles of the substrate and product for a canonical MM mechanism in the real space and can be readily adapted to the more complex mechanisms which will be considered in the following. Since all these species diffuse in the solution their concentration profiles under steady state or quasi-steady state conditions are linear in the appropriate transformed space: $\zeta = r_0(1 - r_0/r)$ for electrodes of radius $r_0$ equivalent to hemispherical ones [12] or $\zeta = r_0 \ln(r/r_0)$ for cylindrical ones [10-12]. As shown in Fig.2b, the effect of diffusion can then be taken into account by considering steady or quasi-steady state diffusion layers of thickness $\delta$ given by:
$$ \delta = r_0 \qquad (9a) $$

or

$$ \delta = r_0 \ln(2\sqrt{Dt_{cal}}/r_0) \qquad (9b) $$

accordingly [10-12], where $t_{cal}$ is the time duration of each calibration step (note that for micro- and nanosensors, $\delta$ does not vary significantly as soon as $t_{cal}$ is larger than a few seconds [10-12], hence the term of “quasi-steady-state” applying to cylindrical diffusion when $2\sqrt{Dt_{cal}} \gg r_0$).

<caption>[Fig.2]</caption>

The flux ($J_S$)$_{\zeta=0}$ at the conducting surface of the enzymatic electrode is null since $S$ is non-electroactive substrate by definition. Conversely, the flux of $S$ towards the solution is (Fig.2b):

$$ J_S^{sol} = D \frac{[S]_{bulk} - [S]_{enz}}{\delta - \zeta_{enz}} \qquad (10) $$

where, for our purpose here, $[S]_{enz}$ represents the $[S]$ value at $\zeta_{enz}$, whose value is defined by the intersection of the two lines figuring the slopes of $S$ concentration profile on either side of the enzymatic layer in the conformal space (Fig.2b). $\zeta_{enz}$ is much smaller than $\delta$ since its value is by definition smaller than that of the thickness of the thin enzymatic coating (see Fig.2). Hence, Eq.(10) can be readily simplified into:

$$ J_S^{sol} = D \frac{[S]_{bulk} - [S]_{enz}}{\delta} \qquad (11) $$

For any electroactive species $P$ and generated by the enzymatic conversion of substrate $S$
and initially absent in the bulk solution, the conservation of fluxes imposes that:

$$ (J_P)_{\zeta=0} - J_P^{sol} = J_S^{sol} - (J_S)_{\zeta=0} \quad (12a) $$

where $(J_S)_{\zeta=0} = 0$ since the substrate is not electroactive and $(J_P)_{\zeta=0} = D \frac{[P]_{enz}}{\zeta_{enz}} = \frac{i}{nFA'}$ while $J_P^{sol} = -D \frac{[P]_{enz}}{\delta - \zeta_{enz}} \approx -D \frac{[P]_{enz}}{\delta}$ where $[P]_{enz}$ is the maximum concentration of P (Fig.2). Hence, since $\delta \gg \zeta_{enz}$, Eq. (12a) readily rewrites as:

$$ J_S^{sol} = D[P]_{enz} \left( \frac{1}{\zeta_{enz}} + \frac{1}{\delta} \right) \approx D \frac{[P]_{enz}}{\zeta_{enz}} = \frac{i}{nFA} \quad (12b) $$

From Eqs.(11) and (12b) it follows that:

$$ i \approx nFA J_S^{sol} \approx nFAD \frac{[S]_{bulk} - [S]_{enz}}{\delta} \quad (13) $$

which establishes that the function $g$ defined in Eq.(4) is then:

$$ g\left(\frac{i}{nFA}\right) = \frac{i\delta}{nFA D} \quad (14) $$

Note that Eq.(14) is general and remains valid for any enzymatic mechanism since it stems only from (i) the diffusional supply of the substrate $S$ to the enzymatic layer and (ii) the electroactivity of the product $P$ at the electrode electroactive surface, both processes occurring in the enzyme-free sections of the electrochemical enzymatic sensor. Conversely, the very nature of the mechanism active in the thin enzyme coating is only acting through the exact analytical dependence of $rate_{enz}$ on $[S]_{enz}$ whose value is related to $[S]_{bulk}$ and to the current intensity through Eq.(13).

Let us then return to the canonical Michaelis-Menten mechanism. Incorporating Eq.(14) into Eq.(5) yields:
$$ \frac{[S]_{\text{bulk}}}{K_M} = \frac{i}{i_{\text{max}}^{\text{MM}}} \left[ \frac{\delta V_{\text{max}}^{\text{enz}}}{D K_M} + \frac{1}{1 - \frac{i}{i_{\text{max}}^{\text{MM}}}} \right] = \frac{i}{i_{\text{max}}^{\text{MM}}} \left[ \sigma_{\text{MM}} + \frac{1}{1 - \frac{i}{i_{\text{max}}^{\text{MM}}}} \right] \quad (15) $$

where $\sigma_{MM} = V_{max}^{enz} \delta/(K_M D)$ and $i_{max}^{MM} = nFA V_{max}$, $V_{max}$ being the maximum rate (in mol/m²/s units) of the sensor (note that for a canonical MM mechanism, $V_{max} = V_{max}^{enz}$ as follows from Eq.(5)). The dimensionless parameter $\sigma_{MM} = V_{max}^{enz} \delta/(K_M D)$ compares reaction and diffusion rates for the system at hand so is, to some extent, related to the square root of the Thiele modulus [22]. Eq.(15) then provides the correct relationship between the current and the substrate bulk concentration. Only this correct relationship is able to provide the kinetic data characterizing sensor under the present conditions. Special care should be taken whenever a calibrated enzymatic electrochemical sensor is aimed to be used under non-steady state conditions as will be shown in a following work [23].

[Fig.3]

Although Eq.(15) is quadratic with respect to the current intensity, it is first order with respect to $[S]_{bulk}$. This property allows to easily construct calibration curves providing $[S]_{bulk}$ as a function of the current for different values of $\sigma_{MM}$ which are shown in Fig.3a. Fig.3b shows the same calibration curves plotted using a double reciprocal scaling. Although the calibration curves in Fig.3a are strongly reminiscent of usual calibration plots, as hinted in the 'Preliminary consideration' section, those in Fig.3b — which are the formal correct equivalents of Lineweaver-Burk plots — show that as soon as $\sigma_{MM}$ values are not negligible 1/i plots vs. 1/[S]$_{bulk}$ are strongly curved except at large $K_M/[S]_{bulk}$ values where a linearity is apparently restored (Fig.3b). Yet, as noted through comparing Fig.3b to Fig.3c (compare the dashed lines in Fig.3c to the dotted lines in Fig.3b) the theoretical limits predicted when $i_{max}^{MM}/i \to \infty$ given in Eq.(16) require extremely large $K_M/[S]_{bulk}$ values to
achieve linear behavior:

$$ \frac{i_{max}^{MM}}{i} \xrightarrow{\frac{K_M}{[S]_{bulk}} \rightarrow \infty} (1 + \sigma_{MM}) \frac{K_M}{[S]_{bulk}} + \frac{1}{1 + \sigma_{MM}} \quad (16) $$

This is a clear proof that the performance and limitations of enzymatic electrochemical sensors cannot be analyzed using approaches developed for homogeneous solutions even for the simplest MM case. This is a very important result of the above analysis. Indeed, to evaluate the performance and limitation of electrochemical bioenzymatic sensors one often uses double reciprocal plots expecting them to be linear for a well-behaving sensor and try to extract slopes (supposedly $K_M/V_{max}^{enz}$ values) and intercepts (supposedly $1/V_{max}^{enz}$) values to characterize their sensors [5-7, 15-18]. Furthermore, Fig.3c shows that unless $\sigma_{MM} = 0$, this frequent but incorrect procedure yields intercepts of $i_{max}^{MM}/i$ that are significantly smaller than unity which is a theoretical non-sense since by definition $i_{max}^{MM}/i \ge 1$. Therefore, except when $\sigma_{MM} = 0$, i.e., when mass transport is excessively fast compared to enzymatic kinetics the usual double reciprocal diagrams provide misleading information. Note also that when $\sigma_{MM} = 0$, (black curves in Fis.3a-c), one has $[S]_{enz} \approx [S]_{bulk}$ so that the corresponding situations are those in which the enzymatic system is almost inactive, the sensors being close to their saturation limit (compare Fig.3a). Note that the same considerations would also apply to other types of (semi)reciprocal plots such as Hanes-Woolf or Eadie-Hofstee analyses [14] if used to treat bioelectrochemical enzymatic sensor data.

However, plotting calibration curve as a function of $[S]_{enz}$ would provide a linear curve in double reciprocal scale since in these case equations are equivalent to the classical Lineweaver-Burk relations. Yet, generally $[S]_{enz}$ is not known, thus preventing construction of these simple to analyze calibration curves.
In fact, the dependence of $[S]_{enz}$ with $[S]_{bulk}$ for non-negligible values of $\sigma_{MM}$ is obtained by combining Eqs.(1), (2) and (11). This yields a quadratic equation which is readily solved analytically to afford:

$$ \frac{[S]_{enz}}{K_M} = \frac{1}{2} \left[ h + \sqrt{h^2 + 4 \frac{[S]_{bulk}}{K_M}} \right], \quad \text{where: } h = \frac{[S]_{bulk}}{K_M} - (1 + \sigma_{MM}) \quad (17) $$

A set of these solutions is shown in Fig.4 for the $\sigma_{MM}$ values already used in Fig.3. All these plots display two interesting limiting behaviors (see the dotted lines in Fig.4 for $\sigma_{MM} = 4$):
(i) a tangent at $[S]_{bulk}/K_M = 0$ whose equation is:

$$ [S]_{bulk}/K_M \to 0: \quad [S]_{enz} \to [S]_{bulk}/(1 + \sigma_{MM}) \quad (18a) $$

and (ii) an asymptote with a unity slope at large $[S]_{bulk}/K_M$ values:

$$ [S]_{bulk}/K_M \to \infty: \quad [S]_{enz} \to [S]_{bulk} - \sigma_{MM}K_M \quad (18b) $$

These two limits in Eqns.(18a,b) and the noticeable curvatures at intermediate $[S]_{bulk}/K_M$ values emphasize again that $[S]_{enz}$ always differs from $[S]_{bulk}$ unless when $\sigma_{MM} \ll 1$, that is when the mass transport rate drastically outperforms that of the enzymatic activity as exemplified by the black curve ($\sigma_{MM} = 0$) in Fig.4. Though, as said above, this corresponds to a highly inefficient enzymatic activity.

[Fig.4]

## 3.2 Bisubstrate ping pong mechanisms (PP and PI)

In this section we would like to consider a bisubstrate ($S$ and $S_2$) ping pong mechanism (Scheme 2) since it is one of the most frequently encountered situations in bioelectrochemical sensors. Two versions of ping pong mechanism will be considered: (i)
one, denoted PP in the following, when $k_b^{cat} = 0$ as more often considered [24] or (ii) in the general case, noted PI since it may involve product inhibition, when $k_b^{cat} > 0$, i.e. when the enzyme is inhibited by the intermediate product Q [14]. It should be noted that the possible additional inhibition by final product P is generally not very efficient since under amperometric conditions P concentration at the electrode surface vicinity is extremely small.

$$E + S \underset{k_b}{\overset{k_f}{\rightleftharpoons}} ES \underset{k_b^{cat}}{\overset{k_f^{cat}}{\rightleftharpoons}} E^* + Q$$

$$E^* + S_2 \underset{k_{b2}}{\overset{k_{f2}}{\rightleftharpoons}} E^*S_2 \underset{k_f^{cat2}}{\overset{k_f^{cat2}}{\rightleftharpoons}} E + P$$

$$P \underset{\text{current}}{\overset{ne^-}{\rightleftharpoons}} S_2 + Y$$

**Scheme 2.** Ping Pong Bi Bi electrochemical mechanisms referred hereinafter to as PP and PI depending on the reversibility of the first enzymatic step: PP when $k_b^{cat} = 0$, or PI for $k_b^{cat} > 0$, i.e., when product Q may inhibit the first step. The global mechanism is detailed in the common situation encountered with oxidases when the final product P electrochemical detection regenerates the co-substrate $S_2$, i.e., generalizing the ubiquitous case where $S_2$ is $O_2$ and P is $H_2O_2$ ($n=-2$).

For bioelectrochemical sensors developed to monitor the concentration of the main substrate of interest, viz., S in Scheme 2, it is advisable to independently maintain the concentration of the co-substrate $S_2$ as constant as possible, a constraint that may be difficult to fulfill. For this reason, many authors favor oxidases whenever possible to detect
and quantify S through its 2e-oxidation involving dioxygen (O₂) as the electron acceptor and yielding H₂O₂ as the final enzymatic product P in Scheme 2. Indeed, H₂O₂ electrochemical oxidation regenerates O₂ which automatically takes care of the above constraint (see bottom equation in Scheme 2). For this reason, in order to simplify the following presentation and focus on the issue of interest in this work we will only consider this ubiquitous case (Scheme 2).

Ping Pong Bi Bi electrochemical mechanisms such as the one detailed in Scheme 2 [24] are often approximated by a canonical Michaelis-Menten MM mechanism (Scheme 1) [5-7]. However, this supposes that the first step of the Scheme 2 is globally irreversible, i.e., $k_b^{cat}[Q] \approx 0$, and also that the second step is infinitely fast relative to the first one. Despite the fact that this approximation is often used we wish to separate in the following the situations depending on whether or not $k_b^{cat}[Q]$ has a negligible kinetic role.

The rate law for PP mechanisms, i.e. in the absence of any product inhibition, is given by [13, 14]:

$$rate_{enz}^{PP} = \frac{V_{max}^{enz}}{1 + \frac{K_{M}}{[S]_{enz}} + \frac{K_{M2}}{[S_{2}]_{enz}}} \quad (19)$$

where:

$$K_M = \frac{k_f^{cat2}(k_f^{cat} + k_b)}{k_f(k_f^{cat} + k_f^{cat2})}, \quad K_{M2} = \frac{k_f^{cat}(k_{b2} + k_f^{cat2})}{k_{f2}(k_f^{cat} + k_f^{cat2})} \quad (20)$$

Rate law (19) involves again the interfacial concentrations of both substrates. However, as discussed above about oxidases when O₂ is the co-reactant S₂ while the electrochemical detection step involves the 2e-oxidation of the formed H₂O₂, S₂ concentration does not suffer significant variations, hence $[S₂]_{enz} \approx [S₂]_{bulk}$. We assume in the following that this
is always the case.

As was noted before, Eqs.(1), (3), (4), (13), (14) take into account the effect of diffusional transport and are independent of the exact mechanism occurring in the thin enzyme layer. Thus, by performing the same derivations as in the MM case but using now the pertinent rate law in Eq.(19) one readily obtains the equation describing calibration curves for PP mechanisms under the present electrochemical conditions:

$$ \frac{[S]_{\text{bulk}}}{K_M} = \frac{i}{i_{\text{max}}^{\text{PP}}} \left\{ \sigma_{\text{PP}} + \frac{1}{\beta \left( 1 - \frac{i}{i_{\text{max}}^{\text{PP}}} \right)} \right\} \quad (21) $$

where:

$$ \beta = 1 + \frac{K_{M2}}{[S_2]_{\text{bulk}}}, \quad i_{\text{max}}^{\text{PP}} = nFA V_{\text{max}}^{\text{enz}}/\beta, \quad \sigma_{\text{PP}} = \sigma_{\text{MM}}/\beta = V_{\text{max}}^{\text{enz}} \delta/(\beta K_M D) \quad (22) $$

The maximum current $i_{\text{max}}^{\text{PP}}$ depends now on the second substrate concentration owing to the presence of the term $\beta \ge 1$ in its denominator albeit the whole sequence remains kinetically controlled by the first irreversible step in Scheme 2 as evidenced by comparing Figs.3 and 5. Nonetheless, the second step may act by lowering the electrochemical current intensity which is now divided by the factor $\beta$ unless $[S_2]_{\text{bulk}} \gg K_{M2}$ (i.e., when $\beta \approx 1$) as evidenced by the above $i_{\text{max}}^{\text{PP}}$ expression in Eq.(22). This indirectly justifies the general use of analyses based on MM mechanistic framework using the kinetic characteristics of the first step although PP mechanisms involve a sequence of two enzymatic steps. Yet it is clear that using this MM pseudo equivalence leads to the determination of apparently smaller $V_{\text{max}}^{\text{enz}}$ values.

More precisely, the smaller $[S_2]_{\text{bulk}}$ with respect to the second Michaelis constant $K_{M2}$, the larger is the decrease of the maximum current compared to that predicted assuming a
canonical MM mechanism involving the same first step when $Q \equiv P$. However, any Lineweaver-Burk plots established for sufficiently large $K_M/[S]_{bulk}$ values will have different significance than they would for a true MM mechanism. This is fully evident upon examining equation (23) that describes the linear asymptote of Eq.(21) at large $K_M/[S]_{bulk}$ values (compare Eq.(16) taking into account that $\sigma_{MM} = \beta\sigma_{PP}$):

$$ \frac{i_{max}^{PP}}{i} \xrightarrow{K_M/[S]_{bulk} \to \infty} \frac{(1+\beta\sigma_{PP})}{\beta} \frac{K_M}{[S]_{bulk}} + \frac{1}{1+\beta\sigma_{PP}} = \frac{(1+\sigma_{MM})}{\beta} \frac{K_M}{[S]_{bulk}} + \frac{1}{1+\sigma_{MM}} \quad (23) $$

This caveat needs to be duly taken into account if the calibration results and their data extracted through Lineweaver-Burk plots need to be used for applications of the same sensor under different experimental conditions.

[Fig.5]

In the case of PI mechanisms, i.e., with inhibition by the intermediate product $Q$, the rate law becomes:

$$ rate_{enz}^{PI} = \frac{V_{max}^{enz}}{1 + \frac{K_M}{[S]_{enz}} + \frac{K_{M2}}{[S_2]_{enz}} + \frac{K_{M2}}{K_{eq}} \frac{[Q]_{enz}}{[S]_{enz}[S_2]_{enz}} + \frac{K_{M2}}{K_{eq2}} \frac{[Q]_{enz}}{[S_2]_{enz}}} \quad (24) $$

where

$$ K_{eq1} = \frac{k_f}{k_r}, \quad K_{eq2} = \frac{k_{cat}}{k_{cb}}, \quad \text{and} \quad K_{eq} = K_{eq1}K_{eq2} = \frac{k_f k_{cat}}{k_r k_{cb}} \quad (25) $$

Let us introduce:

$$ \gamma = \beta + \frac{K_{M2}}{[S_2]_{bulk} K_{eq2}} [Q]_{enz}, \quad i_{max}^{PI} = i_{max}^{MM}/\gamma = nFA V_{max}^{enz}/\gamma \quad (26) $$

Since we assumed that the diffusion coefficients of the substrate and intermediate product are equal, the substrate/product mass conservation equation affords:
$$[Q]_{enz} = [S]_{bulk} - [S]_{enz} = \frac{i \delta}{nFA D} \qquad (27)$$

since this stems from the substrate/product mass conservation equation. By substituting Eq.(27) into (24), taking the limit at $[S]_{enz} \to \infty$ and converting rates into currents (as done when multiplying rates by nFA) one obtains a quadratic equation with respect to $i_{max}^{PI}$ whose solution is:

$$\frac{i_{max}^{PI}}{nFA} = \frac{V_{max}^{enz}}{\gamma} = \frac{1}{2^{\frac{(\beta-1)}{K_{eq2}}} \frac{\delta}{D}} \left( -\beta + \sqrt{\beta^2 + 4 \frac{(\beta-1)}{K_{eq2}} \frac{\delta}{D} V_{max}^{enz}} \right) \qquad (28)$$

The equation of calibration curves under PI mechanism is then given by:

$$\frac{[S]_{bulk}}{K_M} = \frac{i}{i_{max}^{PI}} \left\{ \sigma_{PI} + \frac{1+\eta_1 \frac{i}{i_{max}^{PI}}}{\gamma - \frac{i}{i_{max}^{PI}} \left( \beta + \eta_2 \frac{i}{i_{max}^{PI}} \right)} \right\} \qquad (29)$$

with $\sigma_{PI} = \sigma_{MM}/\gamma = V_{max}^{enz} \delta/(\gamma K_M D)$, $\eta_1 = (\beta - 1)\sigma_{PI}/K_{eq}$, $\eta_2 = \eta_1 K_M K_{eq1}$. This equation is structurally similar to Eqs.(15) and (21), but incorporates additional kinetic terms representing the effect of the partial inhibition of the first step kinetics by the intermediate product Q.

### 3.3 Practical use of equations (15), (21) and (29)

The above theoretical analyses had the merit of elucidating quantitatively through compact analytical formulations the important problem identified in the “Preliminary Considerations” section by evaluating the significant changes introduced by the necessary involvement of the diffusional transport of substrate and product(s) to and from the enzyme thin layer. This establishes that enzymatic kinetic formulations derived for homogeneous solutions need to be adapted to account for the performance of enzymatic electrochemical sensors. However,
equations (15), (21) and (29) depend on a whole set of parameters whose values are generally unknown. Therefore, for practical reasons, it would be useful to provide an access to relationships that involve experimentally available information as much as possible.

The maximum sensor experimental current values $i_{max}$ can be evaluated, or at least reasonably approximated, from calibration curves relating current vs. substrate bulk concentration at high substrate concentrations under steady or quasi-steady state conditions. On the other hand, for a canonical homogeneous Michaelis-Menten mechanism $K_M$ is equal to the substrate concentration value at which the half-maximum activity of the enzyme is observed. Though, this is not the case for electrochemical enzymatic sensors. Eq.(15) shows that under electrochemical conditions the bulk substrate concentration $[S]_{bulk}^{i_{max}/2}$, viz., that at which the sensor current intensity reaches its half-maximum value, $i = i_{max}^{MM}/2$, features not only the $K_M$ value of the immobilized enzyme but includes intrinsic enzymatic (viz., $K_M$, $V_{max}^{enz}$) and transport properties of the system as shown by Eq.(30):

$$[S]_{bulk}^{i_{max}/2} = K_M + \frac{\delta}{2D} V_{max}^{enz} = K_M \left(1 + \frac{\sigma_{MM}}{2}\right) \quad (30)$$

Note that $[S]_{bulk}^{i_{max}/2} = K_M$ is obtained only in the limiting case when $\sigma_{MM} = 0$, that is when the transport of the substrate across the diffusion layer is infinitely fast compared to the enzymatic kinetics taking place at the surface of the electrode.

The explicit definition of $[S]_{bulk}^{i_{max}/2}$ in Eq.(30) allows introducing the dimensionless parameter $\omega_{MM}$:

$$\omega_{MM} = \frac{2\sigma_{MM}}{2+\sigma_{MM}} = \frac{i_{max}^{MM}}{nFA D[S]_{bulk}^{i_{max}/2}/\delta} \quad (31)$$

which plays a role equivalent in meaning to that of $\sigma_{MM}$ by comparing the enzyme kinetics
to the mass transport rate, although this is now performed through involving only experimentally available data, viz. $i_{max}$ and $[S]_{bulk}^{i_{max}/2}$ as well as operational parameters ($A$, $\delta$).

Eqs.(30) and (31) allows rewriting Eq.(15) into Eq.(32) which now only involves factors that can be experimentally accessible since they are readily evaluated from $[S]_{bulk}^{i_{max}/2}$ and $i_{max}^{MM}$ experimental values:

$$ \frac{[S]_{bulk}}{[S]_{bulk}^{i_{max}/2}} = \frac{i}{i_{max}^{MM}} \left[ \omega_{MM} + \frac{1-\omega_{MM}/2}{(1-i/i_{max}^{MM})} \right] \quad (32) $$

Eq.(32) provides a dimensionless “calibration curve” that depends only on $[S]_{bulk}^{i_{max}/2}$ and $i_{max}^{MM}$ and can be plotted without any approximation for any given experimental value of $\omega_{MM}$ whose value is itself available experimentally. A set of such “calibration curves” are shown in Fig.6 in linear and reciprocal scales. Note that whatever is the value of $\omega_{MM}$, the renormalized calibration curves in Fig.6a require a high experimental accuracy to be distinguished when $[S]_{bulk}/[S]_{bulk}^{i_{max}/2} < 1$.

<caption>[Fig.6]</caption>

Along the same logic one may formulate dimensionless “calibration curves” equations for PP mechanism just in terms of the experimentally available parameters $\omega_{PP}$, $[S]_{bulk}^{i_{max}^{PP}/2}$ and $i_{max}^{PP}$ which are defined in Table 1. As before, this results in an equation with an identical structure to that reported for a MM mechanism in Eq.(32):

$$ \frac{[S]_{bulk}}{[S]_{bulk}^{i_{max}^{PP}/2}} = \frac{i}{i_{max}^{PP}} \left[ \omega_{PP} + \frac{1-\omega_{PP}/2}{(1-i/i_{max}^{PP})} \right] \quad (33) $$

It follows that in practice the MM and PP mechanisms cannot be distinguished on the basis
of their calibration curves alone, although the significance of the parameters extracted from their quantitative processing differs profoundly at the level of enzymatic kinetics. This justifies that the PP mechanisms can be treated as simple MM mechanisms even if it is necessary to realize that the analysis now provides apparent values ($V_{max}^{enz}$)app and $K_{M}^{app}$ values that differ from the effective $V_{max}^{enz}$ and $K_{M}$. Indeed, one has then ($V_{max}^{enz}$)app = $V_{max}^{enz}/\beta$ and $K_{M}^{app} = K_{M}/\beta$ (see Eq.(22) or Table 1 for the definition of $\beta$).

<caption>[Table 1]</caption>

For the case of PI mechanisms the resulting equation has structure similar to those in Eqs.(32) and (33), see Eq.(34), but due to the kinetic competition between the direct (i.e. uninhibited) mechanism and its inhibition branch two additional competition parameters $\eta_1 = (\beta - 1)\sigma_{PI}/K_{eq}$, $\eta_2 = \eta_1 K_M K_{eq1}$ are involved in Eq.(29) although they are generally unknown:

$$ \frac{[S]_{bulk}}{[S]^{i_{max}/2}_{bulk}} = \frac{i}{i_{max}^{PI}} \left[ \sigma_{PI} + \frac{1+\eta_1 i/i_{max}^{PI}}{\gamma - i/i_{max}^{PI}(\beta+\eta_2 i/i_{max}^{PI})} \right] / h(\sigma_{PI}) \quad (34) $$

Note that $[S]^{i_{max}/2}_{bulk} = K_M h(\sigma_{PI})$, where $h(\sigma_{PI})$ is a function of $\sigma_{PI}$ defined in Eq.(35)

$$ h(\sigma_{PI}) = \frac{\sigma_{PI}}{2} + \frac{1}{2} \left(1 + \frac{\eta_1}{2}\right) / \left[\gamma - \frac{1}{2}\left(\beta + \frac{\eta_2}{2}\right)\right] \quad (35) $$

The structure of Eqns.(34) and (35) show that, although they are necessarily different from PP or MM mechanisms, PI mechanisms would be very hard to distinguish. Indeed, they exhibit very similar functional dependence on experimental parameters except for the kinetic influence of the additional parameters $\eta_1$ and $\eta_2$ whose values are difficultly predictable. This extends previous conclusions drawn for fully homogeneous situations [13,
14, 25] and further illustrates the added complexity brought by electrochemical enzyme micro- and nanosensors owing to convolution of diffusional transport terms with experimental currents. Indeed, upon increasing the mechanism complexity (viz., from MM to PP and then to PI) the operational parameters which govern the calibration curves quickly become complicated functions of primary parameters (see Table 1). Except for MM mechanism, these operational parameters depend on more than two primary parameters. It ensues that an infinite number of primary parameter combinations can lead to a same calibration curve. This is not a problem in itself whenever the only purpose of the calibration curves is to establish a correspondence between currents and substrate concentrations under precise experimental conditions involving steady or quasi-steady state diffusion transport from bulk solution. However, this becomes a debatable issue with significant consequences when sensors "calibrated" under these conditions are to be used under different conditions, i.e., for monitoring the same substrates under in vivo or ex vivo conditions [6, 7, 26, 27].

**4. Conclusion**

In this work we established theoretically that Lineweaver-Burk analyses justly used for evaluating enzymatic kinetic properties when the enzymes are homogeneously dispersed in a bulk solution cannot be used straightforwardly for the analysis of electrochemical calibration curves recorded with enzyme-modified electrodes. Analytical equations extending Lineweaver-Burk analysis to the electrochemical conditions under (quasi-)steady state conditions were derived to allow proper analysis of enzymatic electrode calibration curves for Michaelis-Menten enzymatic mechanisms and ping pong bi-substrate ones with
or without inhibition. More complex enzymatic mechanisms can be equally considered
within the framework of the proposed approach. In addition, our analytical equations can
be reformulated to fit the format of other reciprocal plots frequently used in enzymatic
analysis, such as Hanes-Woolf and Eadie-Hofstee plots [14].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported in parts by CNRS, ENS - PSL University, Sorbonne University (UMR 8640 PASTEUR) in Paris and by the National Natural Science Foundation of China (NSFC) in China. The Chinese team acknowledges financial support from the NSFC (Grants 22090050, 22090051, and 21721005). All authors acknowledge the Sino-French IRP CNRS NanoBioCatEchem for the support. CA thanks as well Xiamen University for his Distinguished Visiting Professor position.

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<table><caption>**Table 1.** Definitions of parameters involved in Eqs.(32), (33) and (34) for each of the three mechanisms considered here.</caption><thead><tr><th>Enzymatic mechanism</th><th>σ</th><th>[S]<sup>i<sub>max</sub>/2</sup><sub>bulk</sub></th><th>ω</th><th>i<sub>max</sub></th></tr></thead><tbody><tr><td>MM: Michaelis-Menten,<br>Eq.(32)</td><td>σ<sub>MM</sub> = <sup>V<sub>max</sub><sup>enz</sup>δ</sup>/<sub>K<sub>M</sub>D</sub></td><td>K<sub>M</sub>(1 + <sup>σ<sub>MM</sub></sup>/<sub>2</sub>)</td><td>ω<sub>MM</sub> = <sup>2σ<sub>MM</sub></sup>/<sub>2 + σ<sub>MM</sub></sub></td><td>nFA V<sub>max</sub><sup>enz</sup></td></tr><tr><td>PP: Ping pong Bi Bi,<br>Eq.(33) <sup>(a)</sup></td><td>σ<sub>PP</sub> = <sup>1</sup>/<sub>β</sub> <sup>V<sub>max</sub><sup>enz</sup>δ</sup>/<sub>K<sub>M</sub>D</sub></td><td>K<sub>M</sub>(1 + <sup>σ<sub>PP</sub></sup>/<sub>2</sub>)</td><td>ω<sub>PP</sub> = <sup>2σ<sub>PP</sub></sup>/<sub>2 + σ<sub>PP</sub></sub></td><td>nFA (V<sub>max</sub><sup>enz</sup>/β)</td></tr><tr><td>PI: Ping pong Bi Bi<br>with inhibition by<br>intermediate product,<br>Eq.(34) <sup>(b,c)</sup></td><td>σ<sub>PI</sub> = <sup>1</sup>/<sub>γ</sub> <sup>V<sub>max</sub><sup>enz</sup>δ</sup>/<sub>K<sub>M</sub>D</sub></td><td>K<sub>M</sub> h(σ<sub>PI</sub>)</td><td>ω<sub>PI</sub> = <sup>σ<sub>PI</sub></sup>/<sub>h(σ<sub>PI</sub>)</sub></td><td>nFA (V<sub>max</sub><sup>enz</sup>/γ)</td></tr></tbody></table>

(a) $\beta = 1 + K_{M2}/[S_2]_{bulk}$; (b) $\gamma = 2 \frac{(\beta-1)}{K_{eq2}} \frac{\delta}{D} / \left( -\frac{\beta}{V_{max}^{enz}} + \sqrt{\left(\frac{\beta}{V_{max}^{enz}}\right)^2 + 4 \frac{(\beta-1)}{K_{eq2}V_{max}^{enz}} \frac{\delta}{D}} \right)$; (c) $h(\sigma_{PI}) = \frac{\sigma_{PI}}{2} + \frac{(1 + \eta_1)}{2} / \left(\beta + \eta_2\right)$,

with $\eta_1 = (\beta - 1)\sigma_{PI}/K_{eq}$, $\eta_2 = \eta_1 K_M K_{eq1}$.
**Figure captions:**

**Fig.1.** Schematic illustration of sensor working principle: S is the non-electroactive substrate analyte, P is the electroactive enzymatic product (co-factors are not represented for sake of simplicity and generality).

**Fig.2.** Schematic representation of the concentration profiles of the involved species at cylindrical or hemispherical enzymatic micro- or nanoelectrodes in the real (a) and conformal (b) spaces [10-12]. The scheme is not drawn to scale for clarity (viz., in fact: $\zeta_{enz} \ll \delta$ and $[P]_{enz} \ll [S]_{bulk}$ because conservation of matter implies that S is mostly converted into the final product Y (see Fig.1 and Scheme 1). See Eqns (9a,b) for the definition of the diffusion layer thickness $\delta$.

**Fig.3.** Canonical Michaelis-Menten mechanism. Theoretical calibration curves evaluated for enzymatic electrode according to Eq.(15) in (a) linear and (b,c) double reciprocal scales (dimensionless equivalents of Lineweaver-Burk plots) for several $\sigma_{MM} = V_{max}^{enz} \delta/(K_M D)$ values. In (b) the colored dotted lines are the linear regressions performed on the segments delimited by the squares on each curve ($R^2 \ge 0.999$). In (c) the same data as in (b) together with their linear regression segments are reproduced, while the gray dashed lines correspond to the limits given by Eq.(16).

**Fig.4.** Canonical Michaelis-Menten mechanism. Variations of $[S]_{enz}/K_M$ as a function of $[S]_{bulk}/K_M$ for several $\sigma$ values labelled on each curve (identical to those in Fig.3). The red dotted lines are the limiting cases given by (18a) and (18b) for $\sigma_{MM} = 4$. The black line ($\sigma_{MM} = 0$) is a line with a unit slope.
**Fig.5.** Ping pong Bi Bi (PP) mechanism calibration curves evaluated based on Eq.(25) for $\sigma_{PP} = 1$ (A) and 4 (B) and $\beta = 1, 2, 4, 10$ in real (a) and double reciprocal (b) scales.

**Fig.6.** Simulated calibration curves plotted using normalized parameters based on experimentally accessible data $i_{max}$ and $[S]_{bulk}^{i_{max}/2}$ for canonical Michaelis-Menten (Eq.32) and ping pong Bi Bi (PP) (Eq.33) mechanisms. (a) Linear scale; (b) double reciprocal scale. The color code characterizing each $\omega$ value (see Eq.(32)) is the same in (a) and (b). See Table 1 for the specific definitions of $i_{max}$, $[S]_{bulk}^{i_{max}/2}$ and $\omega$ according to the MM or PP mechanism.
<figure><img src="image_2.png" /><figcaption>Fig.1.</figcaption></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Fig.2.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig.3.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig.4.</figcaption></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig.5.</figcaption></figure>
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig.6.</figcaption></figure>