Electrochemical Impedance Spectroscopic Analysis of Diffusion-Layer Thickness Distribution Associated with Flagellar Motion of Volvox carteri

Abstract

Although electrochemical methods have been used to relate microbial motility with ionic or redox signals, the evaluation of diffusion-layer variations induced by such motility using electrochemical impedance spectroscopy (EIS) remains limited. Thus, in this study, EIS is used to investigate how the flagellar activity of Volvox carteri modulates the diffusion layer above an electrode. Finite-element simulations and distributed equivalent-circuit modeling are performed for both uniform and non-uniform diffusion-layer thicknesses. Simulations predict that phototactic flagellar convection results in impedance spectra with a slope lower than 45° in the mid-frequency region and a pronounced finite-diffusion bend at low frequencies. These features are validated through experiments involving Volvox-immobilized electrodes under dark and illuminated conditions. Light irradiation reduces the effective diffusion-layer thickness from 2.4 × 10⁻² cm to 6.6 × 10⁻³ cm and introduces a distributed thickness ranging from 4.5 × 10⁻⁴ to 9.0 × 10⁻³ cm, as extracted by transmission-line fitting. These results provide quantitative insights into bio-induced mass-transport modulation and demonstrate the applicability of distributed diffusion models in bioelectrochemical systems.

1. Introduction

Flagellar motility plays a crucial role in the survival, dispersal, and environmental sensing of various microorganisms, including bacteria and algae. The bacterial flagellar motor is a complex molecular machine that converts ion-motive force across the membrane into rotational torque, enabling propulsion in viscous environments.^1–5 Depending on the species and environmental niche, the motor can be driven by protons (H⁺), sodium ions (Na⁺), or potassium ions (K⁺), with distinct structural and functional adaptations.^6–13 Structural studies have revealed the arrangement of stator and rotor components, ion-conducting channels, and conformational changes associated with torque generation.^14–18

The diversity of ion-coupled motors extends to unconventional systems found in extremophiles, such as alkaliphilic Bacillus and Paenibacillus, whose motors are adapted to operate under high pH and specific ion conditions.^10,19,20 Genetic and biochemical investigations have identified key residues and subunits—such as MotP and Pom proteins—that are critical for ion selectivity and torque production.^12,21–23 Comparative studies involving Vibrio species have demonstrated the ability of motors to switch between Na⁺- and H⁺-driven modes through mutational changes in stator proteins.^7,24–26

In addition to bacteria, flagellar motility is central to the behavior of eukaryotic microorganisms such as Volvox and Chlamydomonas. In these organisms, coordinated flagellar beating generates large-scale flows that influence nutrient transport and cellular aggregation.^27–31 The hydrodynamic coupling between the flagella and the surrounding medium can produce microscale convection currents that may alter the diffusion layer adjacent to surfaces.^32–34 Such flows can be triggered or modulated by environmental cues, such as light (phototaxis), oxygen gradients (aerotaxis), and chemical stimuli (chemotaxis and chemokinesis).^2,4,35–38

Electrochemical techniques have been developed to monitor motility-related phenomena in microorganisms. For example, impedance spectroscopy and amperometric detection have been applied to quantify changes in ionic environments or redox-active species near swimming cells.^39,40 In particular, in situ electrochemical systems have enabled the simultaneous measurement of flagellar motion and taxis responses in immobilized algal cells,^39,40 providing a quantitative link between motility and electrochemical signals. Thus, these approaches offer an avenue to study the influence of localized hydrodynamic activity on the mass-transport properties of electrochemical interfaces.

Despite extensive research on flagellar motor mechanics and environmental responses, the direct electrochemical evaluation of diffusion-layer modifications induced by localized convection remains to be reported. Most prior studies have focused on either the biological aspects of motility or steady-state mass transport in quiescent electrolytes. Integrating microbiological motility systems with electrochemical impedance analysis provides a unique platform to probe dynamic changes in the interfacial mass-transport regimes under biologically relevant flow conditions.

Considering these aspects, this study aims to examine the effect of Volvox flagellar activity on the electrochemical impedance characteristics of a planar electrode. By immobilizing Volvox colonies between microelectrodes and monitoring their activity under controlled light and ionic conditions, we clarify how microscale convection alters the diffusion-layer structure. Through equivalent-circuit modeling and simulation of both uniform and non-uniform diffusion-layer thicknesses, we quantitatively assess the relationship between flagellar-induced flow and the resulting impedance spectra.

2. Theory

When the surface state of an electrode is modulated by the applied potential (or current) signal, the modulation introduces a time constant, adding a new locus to the impedance spectrum. The part of the impedance spectrum originating from elementary reactions at the electrode surface is termed the Faradaic impedance. Faradaic impedance can be used to describe various reaction mechanisms, such as desorption of reaction adsorbates, sequential dissolution, and enzymatic reduction. Additionally, the diffusion impedance considered in this study can be expressed in terms of the Faradaic impedance, as follows:

  

\begin{align} Z_{\text{F}} &= R_{\text{ct}} + \frac{RTl}{n^{2}F^{2}c_{\text{Ox}}^{\ast}D_{\text{Ox}}}\left(\cfrac{\tanh \Biggl(l\sqrt{\cfrac{j\omega}{D_{\text{Ox}}}} \Biggr)}{l\sqrt{\cfrac{j\omega}{D_{\text{Ox}}}}} \right) \notag\\ &\quad + \frac{RTl}{n^{2}F^{2}c_{\text{Red}}^{\ast}D_{\text{Red}}}\left(\cfrac{\tanh \Biggl(l\sqrt{\cfrac{j\omega}{D_{\text{Red}}}} \Biggr)}{l\sqrt{\cfrac{j\omega}{D_{\text{Red}}}}} \right) \end{align} (1)

where R_ct is the charge-transfer resistance; n is the number of participating electrons; F is the Faraday constant; $c^{\ast}{}_{\text{Ox}}$ and $c^{\ast}{}_{\text{Red}}$ denote the bulk concentrations of the oxidized and reduced species, respectively; D_Ox and D_Red represent the diffusion coefficients of the oxidized and reduced species, respectively; R is the gas constant; T is the absolute temperature; l is the diffusion-layer thickness; j is the imaginary unit; and ω is the angular frequency.

As measurements were obtained with only oxidized compounds present in the solution, the second term on the right-hand side can be ignored.

3. Experimental

3.1 Culture of Volvox carteri

Volvox carteri was cultured in Standard Volvox Medium (pH 7.5) at 25 °C under continuous white light illumination at 6000 lx. Aeration was provided through a membrane filter. The culture conditions and illumination parameters were identical to those reported previously.²⁷

3.2 Impedance simulations

Impedance simulations were conducted using the calculated Faradaic impedance Z_F. Two cases were considered: one with a uniform diffusion-layer thickness and the other with a non-uniform diffusion-layer thickness, modeled using the corresponding equivalent circuits (described in Section 3.5). Numerical analysis was performed using MATLAB R2006a (MathWorks, USA) over the frequency range from 100 kHz to 10 mHz, with parameters varied as needed.

3.3 Electrochemical impedance measurements

Electrochemical impedance measurements were performed in a three-electrode configuration. The working electrode (rectangle shape, surface area = 5 cm²) consisted of an electrode pair with immobilized Volvox. The incuvated alga was entrapped on the center of the space between the working electrodes by using a polyion complex membrane prepared from poly (L-lysine) (PLL; Aldrich, Mw = 100,000) as a polycation and poly (styrenesulfonate) (PSS; Aldrich, Mw = 70,000). PLL (25 mM (M = mol dm⁻³) monomer unit, 0.5 µL) and PSS (25 mM monomer unit, 0.5 µL) solutions were successively cast on the polyimide surface, followed by additional dropping of the PLL solution (0.5 µL). The plate was rinsed with ultrapure water and dried for 10 min at room temperature.²⁷ The single incubated alga was cast to be entrapped on the polyion complex membrane by using the micropipet. A saturated KCl/Ag/AgCl electrode served as the reference electrode, and a platinum wire was used as the counter electrode. The electrolyte was 1 mM K₃[Fe(CN)₆] with 50 mM Na₂SO₄. Measurements were conducted using a potentiostat (CompactStat, Ivium, The Netherlands) equipped with an impedance analyzer. The measurement cell was identical to that shown in Fig. 1. This setup is identical to the previous study that detected changes in Volvox phototaxis.²⁷ However, this time only one electrode is being used. Impedance was recorded at 0 V over the frequency range of 100 kHz to 10 mHz, with five data points per decade and an AC perturbation amplitude of 10 mV.

Figure 1.

Schematic of the experimental setup and image of the change in diffusion-layer thickness induced by Volvox flagellar movement.

3.4 Illumination conditions during electrochemical measurements

During electrochemical impedance measurements, Volvox carteri was exposed to continuous white-light illumination at 6000 lx, identical to that during culture (Fig. 1). Illumination was provided from above the measurement cell using a halogen lamp, with the incident light passing through a heat-absorbing filter to minimize temperature rise. These conditions were identical to those described previously.²⁷

3.5 Equivalent-circuit fitting

Impedance spectra were fitted to equivalent circuits using ZView (Solartron, UK). For both the uniform and non-uniform diffusion-layer thickness conditions, the equivalent circuits shown in Fig. 2 were employed. Additional numerical analysis was performed using MATLAB R2006a.

Figure 2.

Equivalent circuits: (a) Uniform diffusion-layer thickness (finite-length Warburg element Z_f); (b) Non-uniform diffusion-layer thickness modeled by the transmission line of finite diffusion segments (Z_f1 … Z_fN) in parallel with local constant phase elements (CPEs). R_sol: solution resistance.

4. Results and Discussion

4.1 Simulation results

4.1.1 Uniform diffusion-layer thickness

Figure 3 shows the simulation results for the uniform diffusion-layer thickness case. Assuming that the electrode was not fixed with a baffle, the diffusion-layer thickness was considered to be controlled only by natural convection, with no effect of bioconvection on the electrode (l = 0.05 cm). In the high-frequency range, a capacitive semicircle corresponding to charge-transfer reactions was observed. In the intermediate frequency range (approximately 0.1 Hz) and low-frequency range, a straight line with a slope of 45° was observed, attributable to the Faraday impedance Z_F. This behavior was ascribed to the diffusion-layer thickness being unaffected by biological convection from Volvox, leading to infinite diffusion behavior.

Figure 3.

Simulated Nyquist plots showing the effect of the diffusion-layer thickness (l) on the diffusion impedance: l = 0.050, 0.010, 0.005, and 0.003 cm. A smaller l results in stronger finite diffusion bending toward the real axis at low frequencies. Common kinetic parameters were used across cases.

In contrast, when Volvox was assumed to be fixed to the electrode, the diffusion-layer thickness was set as l = 0.012 cm in the simulation, given that the diffusion-layer thickness was influenced by biological convection from Volvox activity on the electrode surface. A capacitive semicircle was obtained in the high-frequency region, similar to the previous case. Additionally, in the intermediate frequency range, a straight line with a slope of 45° was observed, attributable to Z_F. However, in the low-frequency range (below 0.1 Hz), finite diffusion behavior bending toward the real axis emerged. Furthermore, as the diffusion-layer thickness decreased, i.e., as the influence of biological convection caused by the flagellar motion of Volvox intensified, more pronounced finite diffusion behavior was observed (l = 0.005, 0.003 cm).

These results demonstrate that as the flagellar strength of Volvox increases and biological convection becomes more pronounced, the diffusion-layer thickness decreases, and the impedance behavior of finite diffusion becomes increasingly prominent. The parameters used in the simulations are summarized in Table 1. It is noted that even when an LED light is illuminated on an electrode where Volvox is not fixed, no changes occur due to thermal convection in the diffusion layer.²⁷

Table 1. Parameters used in the simulation.

Parameter	Value	Unit
F	1.00 × 104–1.00 × 107	Hz
R	8.31	J K−1 mol−1
Rsol	3.00 × 102	Ω
T	298	K
Rct	1.00 × 102	Ω
n	1	—
c*	1.00 × 10−3	mol cm−3
T(CPE)	9.00 × 10−3	F s(p−1)
D	8.60 × 10−6	cm2 s−1
A	4.00 × 10−2	cm2
F	9.65 × 104	C mol−1

4.1.2 Non-uniform diffusion-layer thickness

The equivalent circuit shown in Fig. 2b was used to perform impedance simulations for the case with a non-uniform diffusion-layer thickness. The distribution models of the diffusion-layer thickness are shown in Fig. 4a. This figure compares the distribution of the diffusion-layer thickness on the electrode with its average distribution and presents examples of alternative distributions. Figure 4b presents the corresponding simulation results.

Figure 4.

(a) Models of non-uniform diffusion-layer thickness on the electrode. Comparison of a spatially distributed thickness profile (green line) with its arithmetic mean (red line); examples of alternative distribution functions. (b) Impedance simulations using the distribution models in (a). Compared with the uniform layer (same mean thickness), the distributed case exhibits a slope lower than 45° in the intermediate frequency range and more pronounced finite diffusion curvature at low frequencies.

When using the two models in Fig. 4a, a capacitive semicircle was observed in the high-frequency region. In the intermediate to low-frequency range, when no distribution of the diffusion-layer thickness was considered, the behavior was consistent with general finite diffusion: In the intermediate regime, a straight line with a 45° slope, attributable to Z_F, was observed, whereas at low frequencies, the line bent toward the real axis. In contrast, despite having the same total thickness of the diffusion layer, simulations with a distributed diffusion-layer thickness showed diffusion impedance behavior with a slope lower than 45° in the intermediate frequency range. In the low-frequency range, finite diffusion behavior, with the line bending toward the real axis, became more pronounced. This observation indicated that non-uniform diffusion-layer thickness results in behavior different from general finite diffusion.

4.2 Electrochemical impedance measurement

Figure 5 shows Nyquist plots of impedance measured with and without light irradiation using a Volvox-immobilized electrode. The applied DC voltage was 0 V. In the absence of light irradiation, a capacitive semicircle was observed in the high-frequency region, and a straight line with a slope of 45°, attributable to Z_F, was observed in the intermediate frequency region. The line bent slightly toward the real axis in the low-frequency region, but no significant finite diffusion behavior was observed. This suggests that the flagellar movement of Volvox is weak under dark conditions, resulting in only a small influence of biological convection on the electrode. Consequently, the diffusion-layer thickness on the electrode is largely uncontrolled, explaining the absence of prominent finite diffusion behavior.

Figure 5.

Experimental impedance spectra (Nyquist plot) for a Volvox-immobilized electrode at E = 0 V vs. Ag/AgCl in 50 mM Na₂SO₄ with 1 mM K₃[Fe(CN)₆]. Frequency range: 100 kHz–10 mHz; perturbation amplitude: 10 mV; five points/decade. Spectra recorded under light irradiation (phototaxis on, green circles) and in the dark (red circles).

Under light irradiation, a capacitive semicircle was observed in the high-frequency range, and a straight line with a 45° slope, associated with Z_F, was observed in the intermediate frequency range. However, at low frequencies, the finite diffusion behavior (line bending toward the real axis) was more pronounced than that under non-irradiated conditions. This is attributable to the increased flagellar activity of Volvox under light irradiation, which intensifies the influence of biological convection on the electrode, thereby controlling the diffusion-layer thickness and resulting in finite diffusion behavior. It is noted that even when an LED light is illuminated on the electrode where Volvox is not fixed, no changes occur due to thermal convection in the diffusion layer.

4.3 Equivalent-circuit fitting

The equivalent circuit shown in Fig. 2a was used to fit the impedance measurement results in Fig. 5. The results are shown in Fig. S1 of the Supporting Information. The parameter values are summarized in Table S1 of the Supporting Information and Table 2.

Table 2. Fitting parameters with light irradiation.

Parameter	Value	Unit
f	1.00 × 104–1.00 × 107	Hz
R	8.31	J K−1 mol−1
Rsol	5.50 × 10−2	Ω
T	298	K
Rct	9.00 × 10−2	Ω
n	1	—
c*	1.00 × 10−3	mol cm−3
T(CPE)	9.80 × 10−3	F s(p−1)
D	8.60 × 10−6	cm2 s−1
A	6.99 × 10−2	cm2
F	9.65 × 104	C mol−1

The diffusion-layer thickness without and with light irradiation was 2.4 × 10⁻² cm and 6.6 × 10⁻³ cm, respectively. The value measured during light irradiation is calculated from the fitted value assuming no diffusion layer is present. Notably, in the low-frequency region, the tendency of bending toward the real axis under light irradiation differed from typical finite diffusion behavior. This difference is attributable to the distribution of the diffusion-layer thickness. To consider this effect, the equivalent circuit shown in Fig. 2b was used for fitting. The fitting results are shown in Fig. 6a, and the parameters used for fitting are listed in Table 2. The parameters for the diffusion-layer thickness were obtained from the distribution values shown in Fig. 6b.

Figure 6.

Fitting with a distributed diffusion layer under illumination. (a) Nyquist plot comparing data with uniform-layer and distributed-layer fits; (b) Reconstructed distribution of diffusion-layer thickness l across the electrode surface. Green line: single uniform thickness (l = 6.6 × 10⁻³ cm); silver line: spatial distribution (l ≈ 4.5 × 10⁻⁴ to 9.0 × 10⁻³ cm).

Using this distribution model, excellent fitting results were obtained. The distribution of the diffusion-layer thickness ranged from 4.5 × 10⁻⁴ cm to 9.0 × 10⁻³ cm across the electrode surface. Considering that the diffusion layer thickness when not exposed to light is 2.4 × 10⁻² cm, the diffusion layer thickness on the electrode closer to the Volvox when exposed to light is two orders of magnitude smaller. This analysis facilitated the comprehensive evaluation of the distribution of the diffusion layer on the electrode.

5. Conclusions

EIS was used to investigate how the flagellar activity of Volvox carteri modulates the diffusion layer above an electrode. Finite-element simulations and distributed equivalent-circuit modeling were performed for both uniform and non-uniform diffusion-layer thicknesses. Flagellar motion of Volvox dynamically modulates the diffusion layer, with the variations detectable as characteristic changes in electrochemical impedance spectra. Distribution-incorporating equivalent circuits capture the spatial heterogeneity of diffusion-layer thickness and enable quantitative extraction of thickness distributions. These findings can advance the understanding of bio-induced mass transport and promote the development of impedance-based biosensors leveraging motile microorganisms.

Data Availability Statement

The data that support the findings of this study are openly available under the terms of the designated Creative Commons License in J-STAGE Data at https://doi.org/10.50892/data.electrochemistry.30228937.

• 1) Although electrochemical methods have been used to relate microbial motility with ionic or redox signals, the evaluation of diffusion-layer variations induced by such motility using electrochemical impedance spectroscopy (EIS) remains limited. Thus, in this study, EIS is used to investigate how the flagellar activity of Volvox carteri modulates the diffusion layer above an electrode. Finite-element simulations and distributed equivalent-circuit modeling are performed for both uniform and non-uniform diffusion-layer thicknesses. Simulations predict that phototactic flagellar convection results in impedance spectra with a slope lower than 45° in the mid-frequency region and a pronounced finite-diffusion bend at low frequencies. These features are validated through experiments involving Volvox-immobilized electrodes under dark and illuminated conditions. Light irradiation reduces the effective diffusion-layer thickness from 2.4 × 10⁻² cm to 6.6 × 10⁻³ cm and introduces a distributed thickness ranging from 4.5 × 10⁻⁴ to 9.0 × 10⁻³ cm, as extracted by transmission-line fitting. These results provide quantitative insights into bio-induced mass-transport modulation and demonstrate the applicability of distributed diffusion models in bioelectrochemical systems.

CRediT Authorship Contribution Statement

Isao Shitanda: Conceptualization (Lead), Data curation (Lead), Formal analysis (Lead), Supervision (Lead), Writing – original draft (Lead), Writing – review & editing (Lead)

Koji Ishizaki: Data curation (Lead), Formal analysis (Lead), Investigation (Lead)

Hikari Watanabe: Supervision (Lead)

Masayuki Itagaki: Formal analysis (Equal), Supervision (Lead)

Conflict of Interest

The authors declare no conflict of interest in the manuscript.

Footnotes

I. Shitanda and K. Ishizaki: Equal Contribution

I. Shitanda, H. Watanabe, and M. Itagaki: ECSJ Active Members

M. Itagaki: ECSJ Fellow

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