# Modeling the detection of transient vesicular exocytotic release from single cells with cylindrical enzymatic nanoelectrodes

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

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To cite this version:

Reina Dannaoui, Irina Svir, Wei-Hua Huang, Christian Amatore, Alexander Oleinick. Modeling the detection of transient vesicular exocytotic release from single cells with cylindrical enzymatic nanoelectrodes. Electrochimica Acta, 2024, 508, pp.145214. 10.1016/j.electacta.2024.145214. hal-05235188

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# Modeling the detection of transient vesicular exocytotic release from single cells with cylindrical enzymatic nanoelectrodes

Reina Dannaoui<sup>a</sup>, Irina Svir<sup>a</sup>, Wei-Hua Huang<sup>b</sup>, Christian Amatore<sup>a,c,*</sup>, Alexander Oleinick<sup>a,*</sup>

## ARTICLE INFO

### Keywords:

Vesicular exocytosis

Single cell amperometry

Cylindrical nanoelectrode

Kinetic modelling

Enzymatic nanoelectrode

Michaelis-Menten related mechanisms

## ABSTRACT

Amperometric electrochemical detection has proven to be an important versatile tool for studying transient single exocytotic events in tissues and single cells. Meanwhile, advances in electrode materials and micro-fabrication have enabled the routine fabrication of submicrometric electrochemical sensors for in vivo or in vitro analysis of single cells or organelles. In this context, cylindrical nanoelectrodes offer important advantages because their sensing elements can be driven across biological membranes or into intercellular clefts while preserving the biological processes being studied. However, their dimensions and curvatures are comparable to those of the releasing vesicles that deliver their stored material into the extracellular environment. The electrochemical currents are then determined not only by the kinetics of biological release, which is the main objective of monitoring, but also by the intricate diffusion patterns of the analytes in the nanoelectrode-cell space and the kinetics of their electrochemical detection. This situation becomes even more complex when the analytes are not directly electroactive on the electrode surface, but require mediation by an enzymatic process, preventing reliance on classical approaches. Consequently, a theoretical model has been developed to determine and quantify the consequences of these limitations on the electrochemical measurement of transient release processes in quantitative and kinetic terms.

## 1. Introduction

Understanding the mechanism of vesicular exocytotic release of biologically active mediators is a very important biological issue because this phenomenon is involved in many vital processes in cells and organisms, such as neuronal communication via chemical synapses [1], secretion of proteolytic enzymes [2], neurohumoral regulation [3], etc. In this context, the direct amperometric detection of electroactive mediators at microelectrodes has proven to be a versatile tool to study transient individual exocytotic events in tissues and at the level of a single cell. To rationalize these data and to extract kinetic information from such amperometric detection at single cells, physicomathematical descriptions are needed [4-13]. The need for such physicomathematical models has increased with recent advances in electrode materials and micro/nanofabrications, which have allowed the routine fabrication of electrochemical sensors with sub-micrometric dimensions for in vivo or in vitro single cell or single organelle analysis [14-16].

In this context, cylindrical nanoelectrodes have proven to be

exceptionally advantageous because their sensing elements can be driven across biological membranes or inside synaptic clefts between cells [17] while preserving the homeostatic processes of the biological entities under investigation [18]. In this case, however, the dimensions and curvatures of the nanoelectrodes become comparable to those of releasing vesicles that deliver their stored mediators into the extracellular environment. As a result, the electrochemical currents that characterize any analytes' detection are then a priori determined not only by the kinetics of the biological release events, but also by the mediator diffusion patterns in the electrode-cell space as well as by the kinetics of its electrochemical detection. Therefore, unless the electrochemical reactions are sufficiently rapid to be diffusion limited and the diffusional time-of-flight across the extracellular space are negligible compared to the release durations, as was the case in previous research [11,17,19], the kinetics of the electrochemical current cannot accurately reproduce the release flux. This built-in situation is even more complicated when the analytes are not directly electroactive on the electrode surface, so that their detection requires mediation by an enzymatic process. From

<footnote><sup>a</sup> PASTEUR, Département de Chimie, Ecole normale supérieure, PSL Université, Sorbonne Université, CNRS, 24 rue Lhomond, Paris, 75005, France</footnote>

<footnote><sup>b</sup> College of Chemistry and Molecular Sciences, Wuhan University, Wuhan, 430072, PR China</footnote>

<footnote><sup>c</sup> State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, PR China</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: christian.amatore@ens.psl.eu (C. Amatore), oleksandr.oliynyk@ens.psl.eu (A. Oleinick).</footnote>
an analytical point of view, it is therefore necessary to identify and quantify what consequences (if any) these possible limitations have and how they affect the electrochemical measurements of the release processes in quantitative (i.e. collection efficiencies) and kinetic terms (i.e. relevance of the electrochemical currents to the released fluxes).

For these reasons, we wish to investigate and report below the peculiarities imposed by the interweaving of diffusion transport with biological and electrochemical reactivities in such systems and their consequences on the monitoring of brief and intense vesicular exocytotic events by a cylindrical nanoelectrode resting parallel to the surface of the cell membrane at a submicrometric distance. This objective is achieved through the theoretical modeling of the whole system formed by a submicrometric intracellular vesicle releasing its contents through a nanoscale fusion pore, when the electrochemical detection is either direct and fast or mediated by a thin layer of enzyme deposited directly on the surface of a cylindrical nanoelectrode.

## 2. Model

We consider an archetypal situation for monitoring neurotransmitter release with an enzymatic cylindrical nanoelectrode. The electrode is assumed to be positioned at distance *d* parallel to the cell surface, the vesicle and fusion pore are assumed to be aligned with the electrode (i.e. the vesicle symmetry axis passing through the pore center is perpendicular to the cylinder symmetry axis, see Fig. 1 for a sketch of lateral view and figures below for 3D view), unless otherwise stated. The nanoelectrode is assumed to be pressed against the cell which implies that its surface is located approximately at a distance *d* of ca. 100 nm from the surface of the cell bilipidic membrane due to the presence of the cell integral proteins and glycocalyx layers on the surface of cell membranes [20,21].

We assume that release is only driven by the strong diffusional dispersion of the analyte in the extracellular cell-electrode gap coupled to its consumption at the electrode surface. In fact, diffusion is considered as a sole mass transport mode in both compartments, proceeding through site-hopping in the intravesicular matrix, or classically in the extracellular space, with diffusion coefficients $D_S^{ves}$ and $D_S (\gg D_S^{ves})$, respectively [22]. For simplicity, we assume that the fusion pore, which ultimately connects the intravesicular compartment to the extracellular cell-electrode cleft, opens instantaneously. It would be possible to include the dynamics of pore opening through a set of specific boundary condition [6-9,11,12,17,19], but we want to focus hereafter on the main issue that stimulated this work, namely the analysis of the complex interplay between diffusion patterns and the kinetics of enzyme-mediated detection of the released species. Therefore, we assume that a vesicle with radius $r_{ves}$ instantly opens a fusion pore of radius $r_{pore}$ at time $t = 0$ through which a substrate *S* is released.

Two kind of electrodes are considered below: (i) a conventional cylindrical electrode of radius $r_0$ allowing a fast substrate electrooxidation at its surface (as it occurs, e.g., when monitoring catecholamines release with carbon electrode [4]), called hereafter as an 'ideal' electrode; (ii) a modified electrode involving a redox oxidase-mediated conversion of the analyte by a very thin enzymatic layer of thickness $\mu \ll r_0$ deposited at the electrode surface followed by an amperometric oxidation of the enzymatic product (i.e., $H_2O_2$) at the electrode surface. Note that the choice of a redox oxidase and a quantitative oxidative detection of $H_2O_2$ allows the assumption that the co-substrate ($O_2$) is constantly regenerated, so its concentration is in steady state and apparent kinetic constants can be used (see Scheme 1). In both cases, the length of the nanoelectrodes is assumed to be much larger than the fusion pore diameter and any other relevant dimension, so that no edge effects are taken into account.

For enzyme modified electrodes, a simple Michaelis-Menten mechanism (Scheme 1) is often assumed for simplification and analysis of experimental results. See, e.g., [23-25] for glutamate sensors involving a still unknown complex mechanism. Moreover, we have previously found that, unless identified enzyme deactivation pathways are involved, the kinetic consequences around a substrate concentration of interest are identical when using locally fitted gross parameter values, while the true mechanism remains unknown anyway [26]. For this reason, we assume in the following that the mediated mechanism in the thin enzymatic layer can be represented by that in Scheme 1 and obeys the system of defined parameters in Eq. (1) [26].

According to Scheme 1, the enzyme conversion rate depends on the following parameters:

$$v_0 = \frac{V_{max}^{enz} [S]_{r_μ}}{K_M + [S]_{r_μ}}, \quad K_M = \frac{k_{cat} + k_r}{k_f}, \quad (1)$$

where $V_{max}^{enz}$ is the maximum enzyme conversion rate at infinite substrate concentration; $K_M$ is the Michaelis-Menten constant; $[S]_{r_μ}$ is the substrate concentration at $r_μ$ enforced by the enzymatic activity and not that at the fusion pore exit as established in our previous contribution relative to the detection from a bulk solution of the substrate [26]. It is important to understand that it is considerably lower than that stored in the vesicle, i.

$$E + S \underset{k_r}{\overset{k_f}{\rightleftharpoons}} ES \underset{k_{cat}}{\overset{k_{cat}}{\rightleftharpoons}} E + P \\ P \pm ne^- \rightarrow Y$$

**Scheme 1.** Principle of the detection of a non-electroactive analyte (S) through its enzymatic conversion into its electroactive product (P).

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic cross-section of the system geometric configuration in the symmetry plane defined by the orthogonal symmetry axes of the vesicle and nanoelectrode illustrating (a) the detection principle and (b) summarizing the characteristic dimensions (note that in simulations below we consider that $r_μ = r_0 + \mu \approx r_0$). Compare Fig. 6 for a 3D view with cross section perpendicular to the nanoelectrode axis and passing through the vesicle axis.</figcaption></figure>
e., ca. a few 0.1 M.

Under these conditions it is possible to accurately represent the involvement of the enzymatic layer via an apparent boundary condition at $r_μ$ (dashed vertical line in Fig. 2) involving $[S]_{r_μ}$ and $[P]_{r_μ}$, the substrate and product concentrations at $r_μ$ respectively. The flux conservation of substrate and product concentrations inside and just outside the enzymatic layer provides then [26]:

$$J_S^+ = v_0 \qquad (2)$$

$$J_p^+ = -v_0 + J_p^- \qquad (3)$$

where $J_X^-, J_X^+$ are the fluxes of species X at $r_μ$ towards the electrode or bulk, respectively (Fig. 2). Note that we do not need to consider the mass transport of species Y (Fig. 1), since all the quantities of interest can be evaluated based on the distributions of species S and P.

The physicomathematical formulation of the model requires considering the diffusional mass transport equations prevailing inside the vesicle and in the extracellular space:

$$\frac{\partial[S]}{\partial t} = D_S^k \Delta[S] \qquad (4)$$

$$\frac{\partial[P]}{\partial t} = D_p^k \Delta[P] \qquad (5)$$

(where $[P]$ is the concentrations of product; $D_S^k$, $D_p^k$ are the substrate and product diffusion coefficients within the intravesicular matrix or in solution, i.e. $D_S^{ves}$, $D_p^{ves}$ or $D_S$, $D_p$, correspondingly; $\Delta = (\partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2)$ is the Laplacian operator in cartesian 3D space) associated to the initial and boundary conditions:

$$t = 0: \text{ solution}[S] = 0; [P] = 0 \qquad (6)$$

$$\text{vesicle}[S] = [S]_0^{ves}; [P] = 0 \qquad (7)$$

$t > 0$: vesicle matrix/solution interface

$$J_S^{ves} = J_S^{sol}; J_p^{ves} = J_p^{sol} \qquad (8)$$

$$[S]_{pore}^{ves} = \kappa_S [S]_{pore}^{sol} \cdot [P]_{pore}^{ves} = 0 \qquad (9)$$

where $\kappa_S$ is the partition coefficients of the substrate between the vesicle and the solution on each side of the fusion pore. For simplification, in order to reduce the number of parameters $\kappa_S$ is set equal to one in the

<figure><img src="image_5.png" /><figcaption>Fig. 2. Schematic illustration of substrate and product concentrations profiles as well as their fluxes at the interface between the enzymatic layer and solution in the symmetry plane of the system in the region close to the electrode. r is the curvilinear coordinate along one of the many diffusion trajectories connecting the fusion pore to the points of the electrode where the substrate is converted by the enzyme (see Fig. 6). Note that the thickness of the enzyme layer (light green shade) is greatly magnified for the sake of clarity.</figcaption></figure>

numerical results disclosed hereafter.

$$\text{Electrode}: J_S^+ = v_0; \qquad (10)$$

$$J_p^+ = -v_0 + J_p^- \qquad (11)$$

Cell/vesicle membranes and symmetry planes (see below):

$$J_S = 0; J_p = 0; \qquad (12)$$

$$\text{Infinity}: [S] \to 0; [P] \to 0. \qquad (13)$$

where $[S]_0^{ves}$ is the initial substrate concentration inside the vesicle; $J_X^{ves}$, $J_X^{sol}$, $[S]_{pore}^{ves}$, $[S]_{pore}^{sol}$ are the local fluxes ($J_X = D_X \partial[X]/\partial \vec{n}$, $\vec{n}$ being a unit normal vector to the surface boundary of interest) and local concentration of species X ($X = S$ or P) at the mouth of the pore at the interface between vesicle matrix and extracellular solution, respectively.

Note that when the vertical axis of the vesicle in Fig. 1 intersects perpendicularly the horizontal axis of the nanoelectrode, as we consider hereafter, two planes of symmetry coexist, each containing the vertical axis of the vesicle, one including also the axis of the electrode while the other one is orthogonal to it. The problem can therefore be solved in a single quadrant of the 3D space delimited by these two planes of symmetry on whose surfaces a zero flux condition is imposed for each species as stated in the above boundary conditions set. Hence, solving the problem needs to consider a quarter of the 3D space, which reduces computation times and memory occupation by one fourth which is a notable improvement for numerical solutions of partial differential equations in three dimensions. In the most general case, namely when the axes of the vesicle and the electrode do not intersect, the system has only one plane of symmetry passing through the axis of the vesicle and being perpendicular to the axis of the electrode. However, in the following we therefore favored considering the case involving the two symmetry planes, which stipulates specifically that the vesicle axis passing through the fusion pore intersects the cylindrical nanoelectrode axis. Nevertheless, the results presented in the Supporting Information demonstrate that the general conclusions derived hereafter remain qualitatively consistent even if vesicle and electrode axes do not intersect.

The current at the electrode is then:

$$i = nFD \int_{A} \frac{\partial[P]}{\partial \vec{n}} dA \qquad (14)$$

where the integration is performed over the whole electrode surface area $A = 2\pi r_0 L$, where $r_0$ is the nanoelectrode radius and L its length.

All simulations presented hereafter were performed in Comsol Multiphysics 6.2 [27]. The following parameters values have been used in simulations unless otherwise stated: $D_S = 5 \times 10^{-10} \text{ m}^2/\text{s}$, $D_p = 1.5 \times 10^{-9} \text{ m}^2/\text{s}$, $D_S^{ves} = 5 \times 10^{-11} \text{ m}^2/\text{s}$, $D_p^{ves} = 0$, $r_0 = 250 \text{ nm}$, $L = 20 \text{ µm}$, $r_{ves} = 60 \text{ nm}$, $r_{pore} = 10 \text{ nm}$, $d = 100 \text{ nm}$, $\mu = 10 \text{ nm}$, $[S]_0^{ves} = 0.3 \text{ M}$, $\kappa_S = \kappa_p = 1$, $n = 2$. $V_{max}$ and $K_M$ values are indicated for each simulation shown. The value of $D_S^{ves}$ was selected arbitrarily in order to ensure its coherence with the half-time widths of amperometric spikes monitored during vesicular glutamate release events by neuronal varicosities, as published in a previous report [23]. Note that this value is ten times larger than the ones reported for monoamines in large dense core vesicles [22,28].

## 3. Results and discussion

Effect of the electrode bulk and activity on the release. Firstly, we would like to address a crucial point regarding the impact of the electrode presence and activity on the kinetic efflux rate and, consequently, on the relationship between the actual and monitored release processes. When the electrode is not active its bulk can be thought as creating a geometric
obstacle slowing down the release of the substrate or, conversely, when active, as favoring the release by consuming the substrate in the electrode-cell space more quickly than diffusion would do in an open 3D environment (compare Fig. 6). Hence, one may suspect that the amperometric sensor alters the process on which it is intended to report. Although this question is rarely raised in a biological context [4], it is not an illusory contemplation since this is precisely what happens during the electrochemical mapping of the concentration map generated by one electrode with another [29] or in SECM according to its operating mode.

To determine whether the positioning of a cylindrical nanoelectrode in proximity to the fusion pore might affect the significance of measurements, a simulation of the vesicle release flux was performed under three discrete scenarios: (i) release from the vesicle into a fully open semi-infinite 3D space (i.e., no electrode present in the system); (ii) release in the presence of an electrochemically inert nanocylinder (i.e., the limiting case of a very inefficient analyte capture), or (iii) of an ideal electrode (i.e., the limiting case of a rapid and efficient analyte capture). The results shown in Fig. 3 prove that, in any case, the fluxes released by the vesicle under each of these situations are largely identical except for negligible differences noticeable only at times when the fluxes become

vanishingly small anyway (around $t > 6$ ms). In fact, the release process is driven by the concentration difference between the average intravesicular concentration of the substrate and its concentration at the mouth of the pore, viz., $J_S^{pore} \propto (|S|_{ves}^{avg} - |S|_{pore})$ [8]. Hence, the results displayed in Fig. 3a,b reveal that at any time when it matters, the substrate concentration in the immediate vicinity of the pore is always negligible compared to that inside of the vesicle. This progressively changes along the release progress when $J_S^{pore} \to 0$, i.e., when anyway the vesicle is already almost emptied (see a discussion in [9]). Therefore, the vesicular release flux is controlled almost entirely by the intravesicular diffusion of the releasable species towards the fusion pore mouth, i.e., $J_S^{pore} \propto |S|_{ves}^{avg}$ [8].

This is easy to understand when one considers that a stationary spherical diffusion layer with a radius of a few tens of nanometers tends to form around the fusion pore within a few microseconds, as would occur for a disk nanoelectrode with the same radius as the fusion pore [30]. On the one hand, such a delay is negligible in view of the much longer duration of release events, which is usually at least a few milliseconds (cf. Fig. 3a) [23,24]. On the other hand, the fate of the substrate

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** (a) Simulated fluxes of the substrate through the fusion pore of a vesicle initially loaded at 300 mM are presented for the following scenarios: (red) ideal electrode (i.e., involving a fast diffusion-limited substrate oxidation at electrode surface) or (black) insulating cylinder is present in the system, $r_0 = 250$ nm with $d = 100$ nm separation from the cell surface; (blue) when there is no electrode in the system. It is noteworthy that all three curves exhibit a similar trend and closely superimpose up to 4 ms. (b) The same fluxes are plotted on a semi-log scale. It is noteworthy that even for an unrealistically small cell-electrode separation of $d = 10$ nm, the data in the inset demonstrate that the effects are almost negligible whenever the released flux values have significant analytical interest. (c) The concentration distribution at $t = 1$ ms for the three cases considered in (a) and (b) is shown. Inside the vesicle, isoconcentration lines correspond to 20, 35, 40, 45, 48, and 49 mM (from top to bottom), while in the extracellular range (dark blue), all concentration values are much lower (see Fig. 6 for precise values in the extracellular cleft). For details regarding the values of the other parameters utilized, please refer to the Model section, specifically the last paragraph.</figcaption></figure>
within this steady-state layer cannot be significantly influenced by
events that occur much further away from the fusion pore, especially
those that occur near an electrode located at ca. 100–200 nm from the
surface of the cell membrane. Indeed, the phenomenology is the same as
in SECM [29]. As a consequence, under usual experimental arrange-
ments [31], the $J_s^{pore}$ value cannot be influenced by the presence of an
electrode (active or inactive) positioned close to the membrane at a
distance that greatly exceeds the dimension of the fusion pore. This fact
is also confirmed by the observation of an exponential decay of the flux
in simulations (compare Fig. 3b) as is expected in the case of free
diffusional release from exocytotic vesicles [6-9]. Nevertheless, to pro-
vide numerical confirmation of the above using a reasoning by absur-
dity, the effect of placing the electrode at an unrealistic distance of 10
nm from the fusion pore opening (which is not achievable in an exper-
imental setting, as previously discussed, but is smaller than usual fusion
pores [8]) was evaluated through simulations. The results presented in
the inset of Fig. 3b confirm this assertion, although it should be
emphasized that the resulting discrepancies remain relatively minor.

**Monitoring release events with an ideal electrode.** The objective of this section is to propose a reference situation in which the collection efficiency is not limited by interfacial kinetics at the electrode surface. For this purpose, we consider an electrode poised at a potential high enough to allow the direct electrooxidation of the analyte of interest at a diffusion-limited regime. Fig. 4a compares the simulated current monitored at such an electrode as a function of time (black curve) with the flux released by an instantaneously open fusion pore (red curve, rescaled by multiplying by $nF$). This shows that the current closely follows the substrate release, except for a very short delay ($< 0.2$ ms) coherent with the substrate time-of-flight over the 100 nm distance from the fusion pore to the electrode surface.

Fig. 4b displays the temporal variations of the time integral of the two currents represented in Fig. 4a to compare the quantities of analytes released and detected in terms of equivalent oxidation charges normalized with respect to $Q_0 = nF(4/3) \pi r_{ves}^3 [S]_{ves}^0$, i.e., to the electric charge equivalent to the releasable quantity initially present in the vesicle. As can be seen, the collected charge, $Q_{chem}/Q_0$, reaches 97% at the end of the plot ($t = 3.2$ ms), viz., when 99% of the initial vesicle content has been released. This indicates that only 2% of the released amount evades its capture by an ideal electrode although the cylindrical nanoelectrode does not subtend a $2\pi$ solid angle when viewed from the fusion pore as would occur using a classical disk electrode placed in

contact with the cell membrane [4].

**Monitoring release events with an enzymatic electrode.** Fig. 5 reports the results of the corresponding simulations for the same nanoelectrode as in Fig. 4 upon assuming now that it is fully covered with a thin sleeve of substrate-specific enzymes that convert it into an electroactive product (see Fig. 1) following a Michaelis-Menten mechanism (Scheme 1). A series of corresponding currents simulated for a variety of $V_{max}$ and $K_M$ values are shown in Fig. 5a (see also Fig. S1 in SI) and compared to the fusion pore flux equivalent and to the ‘ideal’ electrode current (color code is the same as in Fig. 4). As anticipated, sufficiently large $V_{max}$ values, yet still reasonable for oxidoreductases, associated to small $K_M$ ones allow very efficient amperometric detection of the substrate being closer to the case of an ideal electrode. On the contrary, small $V_{max}$ values and large $K_M$ ones lead to sluggish responses and small current peak amplitudes which results in reduced collection efficiencies as evidenced for each set of $V_{max}$ and $K_M$ values in Fig. 5b and Fig. S1 in SI. Notwithstanding these unfavorable circumstances, the collection efficiency does not reach a plateau but rather increases steadily over time. This slow continuous increase indicates that the corresponding substrate fractions that are collected at long times have followed long diffusion trajectories compared to d [32-36]. Nonetheless, as shown below (Figs. 6 and 7), these late collection pathways are very efficient because they involve "fresh" areas of the enzyme cylindrical electrode that are progressively exposed to the substrate flow as the areas lying immediately in front of the fusion pore become increasingly saturated. In other words, lateral diffusion over the nanoelectrode compensates for the low local activity of the saturated enzyme layer areas. This is very different from what would happen if the same enzyme electrode were operating in a bulk substrate solution, where the saturation level of all enzymes would be identical over the whole layer at any given time [26].

**Cooperation between lateral diffusion and enzymatic activity at a cylindrical electrode.** For relatively low $V_{max}$ values and large $K_M$ ones, substrate molecules diffusing directly toward the electrode area closest to the fusion pore reach an area of the enzymatic layer in which enzymes are rapidly prone to operate at a rate close to saturation (see more detailed discussion below). Thus, lateral diffusion provides access to less saturated areas of the enzyme layer where rapid conversion of the substrate to the electroactive product can occur (Fig. 6). More precisely, the diffusion reaction pathways from a point-like source to a cylindrical electrode (Figs. 6Ia and IIa) involve a delicate convolution of two limiting cases: (i) thin-layer diffusion along the generatrix of the

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** (a) Electrochemical amperometric response (black) of an ‘ideal’ cylindrical electrode (250 nm radius) positioned parallel to the cell membrane surface at 100 nm from it. The analyte flux released through the vesicle fusion pore (red) is given in equivalent current units through multiplication by $nF$ to ease the comparison. In both case the electrochemical stoichiometry was $n = 2$ [23]. (b) Detected (black) and released (red) charge fractions with respect to $Q_0 = nF(4/3) \pi r_{ves}^3 [S]_{ves}^0$ the initial vesicle releasable content.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Simulated (a) electrochemical current and (b) collection efficiencies of the enzymatic electrode for $V_{max}^{enz} = 4.8 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$ and various values of $K_M = 0.7, 0.28, 0.14, 0.07, 0.0175$ mM (from bottom to top in each panel). The red and black curves correspond to the substrate release through the pore and detected by an ideal electrode response, respectively. Collection efficiencies are defined as the percentage of collected amount of molecules at any given time vs. the total releasable amount present initially in the vesicle. See also Fig. S1 in SI for a more extensive set of results for $V_{max}^{enz}$ and $K_M$ values.</figcaption></figure>

cylindrical electrode (Figs. 6Ib and IIb); and (ii) lateral diffusion along the curved boundary of the cylinder (Figs. 6Ic and IIc). Both processes cooperate to bypass the saturating areas located in front of the fusion pore, thereby providing higher collection efficiencies. Yet, this is achieved at the cost of increasing the travel time required to reach increasingly distant unsaturated enzyme sites (see Fig. 6; see also Fig. 7). Therefore, the collection currents remain significant compared to what would be predicted based on calibration curves performed for the same sensor in bulk solutions, but their sluggishness increases for enzymes with low $V_{max}^{enz}$ values and large $K_M$ ones.

The pronounced curvature of the lateral pathways (Figs. 6Ic and IIc) contributes to the overall sluggishness by involving extended diffusion trajectories initially oriented towards infinity. Indeed, these trajectories must ultimately deviate from their original course due to the substrate capture by the reaction at the electrode surface, which exerts a significantly stronger diffusional pull than the zero concentration boundary prevailing at infinite. However, these hairpin-like trajectories increase

the times-of-flight, which in turn delay the recovered collection rate, as observed in Fig. 5 (and Fig. S1 in SI). The same phenomenon occurs for ideal electrodes and is the origin of the apparent collection loss of about 2% observed in Fig. 4b at 3.2 ms. However, these effects are magnified for enzymatic electrodes with low $V_{max}^{enz}$ values and large $K_M$ ones. The comparison of the trajectories simulated at the same time ($t = 2 \times 10^{-4}$ s) for two different enzymatic nanoelectrodes with $V_{max}^{enz} = 1.2 \times 10^{-4}$ or $9.6 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$ and the same $K_M = 0.07$ mM value (Fig. 6) perfectly illustrate this point. Interestingly, this difference in trajectories lengths does not affect the intrinsic rate of release as indicated by similar patterns of equiconcentration lines within the inside of the vesicle (Figs. 6Ib,c vs Figs. 6Iib,c). As discussed before, this is due to the fact that the substrate concentration within the cell-electrode gap is considerably smaller than within the vesicle at any time of interest so the release flux, $J_{S}^{pore} \propto (|S|_{ves}^{avg} - |S|_{pore}) \approx |S|_{ves}^{avg}$, depends only on the quantity of substrate within the vesicle at any given time [8].

Fig. 7 displays a series of maps of the local substrate flux, $J_S^+$, at any given position within the enzyme layer (see Fig. 2 for the definition of $J_S^+$) normalized to its maximum value, $J_S^{max}$, at several characteristic times $t$ distributed along the course of a typical exocytotic release event (see color scale of $J_S^+/J_S^{max}$ values on the right side of Fig. 7). This series of maps demonstrates that the thin enzyme layer patch situated immediately below the fusion pore always exhibits the highest substrate flux and operates at a closer proximity to the enzyme saturation limit than its surrounding areas. Consequently, the greater the flux released, the greater the extent to which diffusion lines extend beyond the central zone, to reach areas where enzymes can still operate at a faster rate. However, this kinetic stress is alleviated as the released substrate flux diminishes. Consequently, the sensor surface area involved initially increases with time, attains a maximum value, and then decreases almost in phase with the emptying of the vesicle.

A more quantitative understanding can be achieved by rewriting the flux Eq. (10) at (see Fig. 2 for the definitions) as:

$$J_S^+ = D_S \left( \frac{\partial |S|}{\partial \vec{r}} \right)_{r_S} = v_0 = \frac{V_{max}^{enz} |S|_{r_S}}{K_M + |S|_{r_S}} \quad (15)$$

Given that the lengths of the main trajectories in Figs. 6I and II are commensurate with the thickness, $d$, of the cell-electrode gap, it is reasonable to normalize the curvilinear coordinate along the diffusional trajectories, i.e., $\vec{r}$ in Eq. (15), with respect to $d$ by introducing the dimensionless variable $\vec{\rho} = \vec{r}/d$. Similarly, since $|S|_{r_S} = K_M$ is the substrate concentration at which the enzymatic rate, $v_0$, is halved, it is useful to introduce the dimensionless variable $s = |S|/K_M$ so that Eq. (15) becomes:

$$\left(\frac{\partial s}{\partial \vec{\rho}}\right)_{\rho_{\mu}} = \left(\frac{V_{\max}^{enz}}{K_M} \frac{d}{D_S}\right) \frac{s_{\rho_{\mu}}}{1 + s_{\rho_{\mu}}} \quad (16)$$

where all terms are dimensionless except for those in the bracketed one in the right-hand-side of Eq. (16), thus showing that the flux depends on the dimensionless parameter:

$$\sigma = \frac{V_{\max}^{enz} d}{K_M D_S} \quad (17)$$

$\sigma$ is related to (Da$_{II}$), the Damkohler second number, or to the Thiele modulus, $\Phi$, which allow characterizing the ratio of the reaction rate to the diffusion rate [26,37,38]. For example, in the two cases displayed in Fig. 6, $\sigma$ values are 0.89 (Fig. 6I) and 7.13 (Fig. 6II) showing that the larger $\sigma$ the more efficient is the enzymatic conversion.

## 4. Conclusion

The present theoretical work investigated the influence of complex diffusion patterns and electrochemical reactivity on the monitoring of
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Diffusion patterns during monitoring release event. (a) 3D view of the neurotransmitter diffusion layer and its flux lines; (b,c) isoconcentration (indicated values are in mM) and flux lines in the cross sections along (b) and perpendicular (c) to the electrode axis. Concentration distributions corresponds to *t* = 2 × 10⁻⁴ s; *K*<sub>M</sub> = 0.07 mM as well as (I) *V*<sub>max</sub><sup>enz</sup> = 1.2 × 10⁻⁴ mol m⁻²s⁻¹ and (II) *V*<sub>max</sub><sup>enz</sup> = 9.6 × 10⁻⁴ mol m⁻²s⁻¹. Other parameters are as indicated in the last paragraph of the Model section. Note that highlighted range of concentrations is from 0 (dark blue) to 5 mM (dark red); due to that the vesicle interior appears as uniform in color, yet isoconcentration lines reveal the distribution inside the vesicle.</figcaption></figure>

short and intense vesicular exocytosis events using amperometry. This
was achieved by developing a model for the amperometric monitoring of
analyte transient fluxes released from a submicrometric intracellular
vesicle through its nanoscale fusion pore when detection is performed
with a cylindrical nanoelectrode positioned parallel to the surface of the
cell membrane at a submicron distance. Cylindrical nanoelectrodes were
chosen because of their ability to drive sensing elements in single living
cells across their biological membranes or in tissues while preserving the

biological processes being studied. However, this significant biological
advantage is offset by the need to consider dimensions and curvatures
comparable to those of the releasing vesicles.

These factors prevent the electrochemical detection currents from
being treated as being directly representative of the kinetics of the
release events. Accordingly, the resulting complex physicomathematical
situation was investigated analytically and solved through 3D numerical
approaches for typical experimental cases where the electrochemical
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Normalized flux normal to the electrode surface $J_S^s/J_S^{max}$ showing saturation region of the electrode during a release event when the fusion pore opens instantly. Each snapshot corresponds to a point marked on the curve. The spike and fluxes were simulated with $V_{max}^{emz} = 1.2 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$ at $t = 12, 30, 60, 180$ (peak), 438, 672, 1000 µs. Other parameters are as indicated in the last paragraph of the Model section.</figcaption></figure>

detection is either direct and fast or requires mediation by a thin layer of enzyme on the surface of a cylindrical nanoelectrode.

### CRediT authorship contribution statement

**Reina Dannaoui:** Investigation. Irina Svir: Formal analysis. Wei-Hua Huang: Supervision, Funding acquisition. Christian Amatore: Writing – review & editing, Supervision, Conceptualization. Alexander Oleinick: Writing – review & editing, Supervision, Funding acquisition, Formal analysis, Conceptualization.

### 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. RD acknowledges the support of her Ph.D. thesis by the Doctoral School ED 388 (Chimie-Physique et Chimie Analytique de Paris Centre). The Chinese team acknowledges financial support from the NSFC (Grants 22090050, 22090051, and 21721005). All authors acknowledge the Sino-French IRP CNRS NanoBioCatEchem for its support. CA thanks as well Xia-men University for his Distinguished Visiting Professor position.

### Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.electacta.2024.145214.

### Data availability

No data was used for the research described in the article.

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# Modeling the detection of transient vesicular exocytotic release from single cells with cylindrical enzymatic nanoelectrodes

## Supporting Information

Reina Dannaoui,¹ Irina Svir,¹ Wei-Hua Huang,² Christian Amatore,¹,³,* 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, P.R. China

³ State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, P.R. China

## Content:

1. Fig.S1. Simulated currents and collection efficiencies for enzymatic sensors with various parameters.

2. Simulation results for the case when vesicle position is shifted with respect to the axis of the electrode; Figs.S2, S3.

3. Importance of finding "$V_{max}^{enz}$ and $K_M$ equivalent values" that characterize the enzyme activity in the catalytic layer of a bioanalytical electrode over the range of activity of the sensor.

4. References for SI.

### 1. Simulated currents and collection efficiencies for enzymatic sensors with various parameters

Note that we followed the classical way of presenting the results in Fig.S1 as a function of $V_{max}^{enz}$ and $K_M$, although it could be more convenient to rely on $V_{max}^{enz}$ and $V_{max}^{enz}/K_M$ as independent parameters since they control the limiting enzyme kinetics at large and small substrate concentrations in the thin enzyme layers (see Fig.2) [S1].
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>**Fig.S1.** Simulated electrochemical currents (a) and collection efficiencies (b) of the enzymatic electrode for various values of parameters $V_{max}^{enz}$. (I) 1.2 × 10⁻⁴, (II) 2.4 × 10⁻⁴, (III) 4.8 × 10⁻⁴, (IV) 9.6 × 10⁻⁴, (V) 19.2 × 10⁻⁴ mol m⁻² s⁻¹ and $K_M$ = 0.7, 0.28, 0.14, 0.07, 0.0175 mM (curves from bottom to top in each panel). The red and black curves correspond to the substrate loading through the pore and ideal electrode response, correspondingly. Collection efficiencies are defined as the percentage of collected amount of molecules at any given time vs. the total releasable amount present initially in the vesicle. Other parameters are as indicated in the last paragraph of the Model section in the main text.</figcaption></figure>
## 2. Simulation results for the case when vesicle position is shifted with respect to the axis of the electrode

Fig.S2 shows the results of the simulations when vesicle is shifted with respect to the electrode axis by $x_{shift} = 250$ nm (see Fig.S3) and $V_{max}^{enz} = 9.6 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$. For convenience, as above the efflux of the neurotransmitter and response of an ideal electrode are shown by the red and black lines, correspondingly (same data as in Fig.4 in the main text). The response of the system when simulations performed under identical conditions, but the vesicle is not shifted (i.e. $x_{shift} = 0$, as shown in Fig.6 in the main text), is depicted by gray dashed line. As expected the spike peak is smaller when the vesicle is shifted with respect to the electrode. In addition, the decaying branch is also slower due to increased diffusion path the substrate molecule should underwent before reaching the electrode resulting in a lower collection efficiency as evident from Fig.S2b. Nevertheless, the overall diffusion pattern is similar to the case with aligned to electrode axis vesicle (compare isoconcentration lines inside and outside vesicle in Figs.6 in the main text). This includes the smart use of the sensor surface including the surface opposite to those facing the release site as evidenced by highly curved flux lines presented in Figs.6 in the main text.

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

**Fig.S2.** Simulated current (a) and collection efficiency (b) for the case when vesicle symmetry axis is not aligned with the electrode axis (blue). Vesicle offset is 250 nm as illustrated in Fig.S3a, $V_{max}^{enz} = 9.6 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$. The gray dashed line are the curves corresponding to the same parameters, but without any offset of the vesicle positions with respect to the electrode (see Fig.5 in the main text). The red and black lines correspond to currents of the substrate from the pore (i.e. flux of NT converted into apparent current for the ease of the comparison) and at the surface of an ideal electrode as shown in Fig.1 in the main text. Other parameters are as indicated in the last paragraph of the Model section in the main text.
<figure><img src="image_23.png" /></figure>

**Fig.S3.** Diffusion patterns during monitoring release events when the symmetry axis of the vesicle and the electrode are not aligned: offset 250 nm, $V_{max}^{enz} = 9.6 \times 10^{-4}$ mol m$^{-2}$s$^{-1}$. (a) 3D view of the neurotransmitter diffusion layer ($t = 2.1 \times 10^{-4}$ s) and (b) its flux lines and isoconcentration profiles in the cross section perpendicular to the electrode axis layer. Other parameters are as indicated in the last paragraph of the Model section in the main text.
### 3. Importance of finding "$V_{max}^{enz}$ and $K_M$ equivalent values" that characterize the enzyme activity in the catalytic layer of a bioanalytical electrode

The catalytic properties of enzymes immobilized on a bioanalytical electrode are expected to be different from those of freely diffusing enzymes in homogeneous solution. This well-known problem can be formally solved by calibrating the bioelectrode, provided that special care is taken when using approaches derived from Lineweaver-Burk plots as discussed in a previous publication [S1]. Classical Lineweaver-Burk plots are not adequate because enzymes in a catalytic layer do not respond kinetically and thermodynamically to a bulk concentration but to the substrate concentration that they enforce inside the catalytic layer due to their own activity (see $[S]_{r_μ}$ in Fig.2 in the main text). This value can be drastically smaller than the bulk substrate concentration (see $[S]_{bulk}$ in Fig.2 of the main text) [S1], especially in the case of transient release from a nanometric biological vesicle where one would be tempted to identify $[S]_{bulk}$ with the intravesicular substrate concentration.

In addition, although in the main text, we restricted the development of our model to a classical Michaelis-Menten mechanism (Scheme 1 in the main text). Indeed, our goal was not to solve a specific experimental case, but to highlight the general phenomena created by the coupling between kinetics and diffusion when a bioanalytical electrode monitors a transient release from nanometric entities.

Therefore, especially for oxidoreductases, more kinetic and thermodynamic parameters than the single ($V_{max}^{enz}$, $K_M$) pair considered in the main text need to be taken into account for a proper description of the bioanalytical electrode response. In the practice a set of equivalent or functional values need to be determined around the range of $[S]_{r_μ}$ substrate concentrations of interest [S1].

Furthermore, any realistic calibration of a bioanalytical electrode must be performed under conditions that closely mimic those encountered during the detection of transient release from a point-like source. Any calibration performed in a bulk solution cannot reproduce the consequences of the "spreading" nature of the detection shown in Fig.7 in the main text. One way out of this complex conundrum may be to mimic transient releases near the electrode using iontophoresis, as introduced by Wightman et al. for a different purpose, i.e., to determine the amount of an injected reagent in the brain of a living animal [S2]. In fact, the generation of transient pulses of different substrate concentrations with a nanopipette placed near the enzymatic electrode, as shown in Fig.1B of [S2], is expected to provide the necessary equivalent parameters for the exact conditions encountered.
References:

[S1] R. Dannaoui, X.-K. Yang, W.-H. Huang, I. Svir, C. Amatore, A. Oleinick. Importance of diffusional constraints for the quantitative evaluation of calibration curves of enzymatic micro- and nanoelectrochemical sensors. *Electrochim. Acta* 473 (2024) 143425.

[S2] N.B.Herr, B.M. Kile, R.M. Carelli, R.M. Wightman. Electroosmotic Flow and Its Contribution to Iontophoretic Delivery. *Anal. Chem.* 80 (2008) 8635.