# Quantum bioelectrochemical (QBIOL) software based on point stochastic processes

Simon Grall, Ignacio Madrid, Aramis Dufour, Helen Sands, Masaki Kato, Akira Fujiwara, Soo Hyeon Kim, Arnaud Chovin, Christophe Demaille, Nicolas Clément

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Simon Grall, Ignacio Madrid, Aramis Dufour, Helen Sands, Masaki Kato, et al. Quantum bioelectrochemical (QBIOL) software based on point stochastic processes. Communications Chemistry, 2025, 8 (1), pp.210. 10.1038/s42004-025-01603-1. hal-05405717

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# Quantum bioelectrochemical (QBIOL) software based on point stochastic processes

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Simon Grall©<fnref n="1" />,²✉, Ignacio Madrid², Aramis Dufour², Helen Sands², Masaki Kato³, Akira Fujiwara©⁴, Soo Hyeon Kim², Arnaud Chovin⁵, Christophe Demaille⁵ & Nicolas Clément<fnref n="1" />,²✉

Bioelectrochemistry is crucial for understanding biological functions and driving applications in synthetic biology, healthcare, and catalysis. However, current simulation methods fail to capture both the stochastic nature of molecular motion and electron transfer across the relevant picosecond-to-minute timescales. We present QBIOL, a web-accessible software that integrates molecular dynamics, applied mathematics, GPU programming, and quantum charge transport to address this challenge. QBIOL enables quantitative stochastic electron transfer simulations and has the potential to reproduce numerically any (bio) electrochemical experiments. We illustrate this potential by comparing our simulations with experimental data on the current generated by electrode-attached redox-labeled DNA, or by nanoconfined redox species, in response to a variety of electrical excitation waveforms, configurations of interest in biosensing and catalysis. The adaptable architecture of QBIOL extends to the development of devices for quantum and molecular technologies, positioning our software as a powerful tool for enabling new research in this rapidly evolving field.

Simulations have become essential in many scientific fields, not only for reducing risks and costs but more importantly for modeling and understanding complex phenomena, thus advancing science. Molecular systems attract heavy scientific interests due to their potential applications in bio-electrochemical systems, molecular electronics and quantum devices<fnref n="1" />⁻⁴. The case of bio-electrochemistry is of particular interest as an attractive approach to understand nanoscale biological systems or use bio-mimetic approaches with low energy electron transfer toward catalytic or green energy applications⁵⁻¹⁰. It also has a dual role to play in the frame of the “More than Moore” and “Beyond CMOS” paradigms¹¹, contributing to the diversification of both the nature and scale of nanoelectronic devices. Redox molecules can be seen as nano-objects exhibiting exceptional reproducibility and compact size, they are already employed in biosensing applications¹²⁻¹⁶, and as they can be integrated into electrochemical and biological systems, are envisioned for DNA data storage¹⁷⁻¹⁹ or large scales integrated nanoelectrochemical sensing²⁰,²¹. However, such nano-bio-systems exhibit a complexity (moving molecules, stochastic electron transfer) that is not captured by commercial finite element modeling (FEM) solutions²²⁻²⁴ nor by molecular dynamics (MD) and coarse-grained approaches alone.

MD provides invaluable insights on molecular movements, averaged electron transfer rates and molecular structure configurations, but does not

include intermolecular or metal-molecule electron transfers, especially at timescales that would allow a direct comparison with experimental measurements for most applications²⁵⁻²⁹. Coarse-grained simulations helped greatly in bridging the gap between angtrom- and micrometer-scales descriptions of molecular movements²⁶,²⁷,³⁰⁻³², even managing to bridge coarse-grained and FEM³³. However, they do not provide themselves a description of electron transfers resulting in current signals comparable with experiments. Molecular electronics achieves good agreement between simulations, models and molecular junction experiments³⁴,³⁵, based on averaged positions for the calculations of electronic rates, because molecular motions (or vibrational modes) are typically much faster than electron transfer rates. However, this assumption does not always hold for biomolecular motions, which occur across a wide range of timescale and over large distances³⁶⁻³⁸. As for FEM, it models with great success macroscopic behaviors in terms of concentration profiles and average values³⁹, including currents, but is not able to take into account the stochasticity in electron transfer coupled to molecular movements arising at the nanoscale. As an example, for the past 20 years electrochemical DNA (E-DNA) sensors' measurements have led to the belief that molecular diffusion of tethered DNA was anomalously slow⁴⁰⁻⁴³. The early developments of QBIOL helped to show that this was incorrect, and that the electrochemical response of

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e-mail: sgrall@laas.fr; nclement@iis.u-tokyo.ac.jp

<footnote n="1">¹LAAS, CNRS, Toulouse, France. ²LIMMS, CNRS, Meguro-ku, Tokyo, Japan. ³Graduate School of Chemical Sciences and Engineering, Hokkaido University, Kita-ku, Sapporo, Japan. ⁴Basic Research Laboratories, NTT, Atsugi-shi, Kanagawa, Japan. ⁵ITODYS, 5 Université Paris Cité - CNRS, Paris, France.</footnote>
these systems could be fully explained by a reduced electron transfer rate, itself due the low probability of presence ($\rho$) of the redox head at the electrode surface, associated with low reorganization energy ($\lambda$)$^{44–46}$ and complex energetics related to hydrophilic/hydrophobic interactions at interfaces$^{47}$. This eventually led to the discovery of ballistic Brownian motion for nanoconfined DNA$^{45}$. These results, along with remaining open questions and the general trend of performance demands driving development and device miniaturization for greater sensitivity and richer data, motivated the full-scope development of QBIOL presented in this work, addressing at the same time the need to transition from averaged to stochastic event representations of electrochemical reactions$^{48–51}$.

Since the rise of single-molecule electrochemistry$^{52}$, electrochemical systems started to be studied and modeled with a stochastic approach$^{42,49,53,54}$, with such nanosystems already becoming more and more widespread. Recent works have shown that stochasticity could be used to the advantage of the experimentalist to obtain as much, if not more, information than through traditional macroscale measurements$^{50,51,55}$. However, stochastic simulations remained behind both in terms of timescales and variety of the simulated entities, often limited to restrained timescales and free particles$^{48,56–59}$, with orders of magnitude discrepancies for experiments attempting to describe redox biomolecules$^{42}$.

Here we show that our simulator QBIOL enables the reproduction of macroscale electrochemical currents of various electrochemical addressing techniques on free particles as well as MD-simulated biomolecules from single-electron counting. The accuracy and timescales available allow direct comparison with experimental data. We choose to focus here on (bio-) electrochemical applications, and also show a working extension for quantum dots (QD).

# Results

## QBIOL workflow and targets

QBIOL is a web-accessible electron transfer simulation software that allows to obtain the current, its fluctuations and probability of presence for any molecule/system in liquid or solid state that can be simulated with MD (Fig. 1). We introduce several innovations over previous electrochemistry-based approaches, incorporating accurate electron transfer rates informed by the latest advancements in the rapidly evolving field of bioelectrochemistry$^{44–46}$. This allows to count electrons one by one in complex moving systems, reproducing familiar macroscale measurements directly comparable with experimental data while keeping the full information on the current's fluctuations (Fig. 1).

The default electron transfer rates are based on Marcus-Hush (MH) theory$^{60,61}$; see sections S1 and S2 in the Supplementary Information (SI) for details and default values. QBIOL uses existing MD libraries, such as oxDNA, to obtain the position of the electroactive part of the molecule of

interest over time. Further details are available in sections S3 of the SI. Notably, oxDNA accounts for temperature effects and secondary structures arising from Watson-Crick base pairing$^{30–32}$. However, no existing coarse-grained simulation algorithm can fully capture non-canonical interactions (such as base triples or G-quadruplexes) due to insufficient experimental data for accurate parameterization$^{62}$. Position-dependent electronic couplings and reorganization energies are computed from the MH formalism and used to generate time-dependent probabilities of electron transfer, which, combined with random number generation algorithms, are used to obtain the stochastic simulation$^{63}$ at the picosecond scale. Estimating the evolution of electrochemical currents over minutes with one picosecond resolution is beyond the capabilities of current computers using a conventional “naive” approach, largely because of the computational cost of integrating MH rates and the trials of stochastic processes to simulate per molecule. To overcome this limitation, we employ rejection sampling, GPU parallelization and pre-computed rates, enabling the efficient calculation of electrochemical currents over extended timescales (details in sections S4 and S5). In addition, we overcome the “Markovian limitation”, which prevents massive parallel computation, by noticing that redox molecules typically have a small number of charge states (typically two).

For the sake of illustration, a ferrocene molecule (Fc) is considered here as the electroactive part (or as a single molecule) unless explicitly mentioned. QBIOL uses the position of Fc to compute a probability of electron transfer to/from an electrode with a time resolution of $dt_{MD} = 9.09 \times 10^{-13}$ s ($\approx 1$ ps) and up to several minutes. In practice, a user only has to input experimental conditions, such as the number of electrodes, type of experiment, applied voltages, dimension of the confinement (if any), sequence of the molecule (if DNA is simulated), ionic strength, etc., to obtain a current versus voltage and/or versus time curve, called here “numerical experiment.” Details about the time management, MD and algorithms are available in SI.

The performances of QBIOL in terms of timescales, variety of electrochemical voltammetry experiments and accuracy of electron transfer rates are presented in section S6 with comparisons to previous work on stochastic electrochemical simulations. To illustrate QBIOL's capabilities and performance, we have focused on a few relevant application examples to highlight its significance for instrumentalists, biophysicists, sensor design, and its utility in addressing questions, such as signal-to-noise ratio across various electrochemists' communities. Finally, the above-mentioned algorithms and computing effort can also be leveraged to model accurately quantum systems, as illustrated further in this paper.

## Performance evaluation: bioelectrochemical experiments

We present hereafter a range of electrochemical experiments available in QBIOL, representative of its reliability and versatility for various cases of

**Fig. 1 | QBIOL overview.** QBIOL allows for counting electrons individually based on a given rate equation. It takes into account the position in space of the electroactive center or molecule by leveraging molecular dynamic solutions, such as oxDNA, with the possibility to simulate nanoconfined environments. The progress in GPU computing combined with applied maths engineering allowed numerical experiments, such as cyclic voltammetry, AC voltammetry or chronoamperometry with realistic timescales. The ability to simulate realistically complex systems in confined environments has applications in biosensors development, quantum devices simulation and molecular electronics.

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interest for the community of electrochemists and bioelectrochemists. The exact way the current is obtained is described in section S7.

**Cyclic voltammetry.** Cyclic voltammetry (CV) has been the primary tool for electrochemists for nearly a century, allowing the extensive characterization of redox reactions as well as quantitative sensing through electroactive molecule measurements. If other more advanced addressing techniques are today used to circumvent some of its limitations, CV measurements remain a cornerstone of electrochemistry.

Figure 2 shows, as a benchmark, a comparison of the CV behavior of freely moving particles within a nanogap ranging from $v = 0.01 \text{ V.s}^{-1}$ to $10^{10} \text{ V.s}^{-1}$, with one of two walls of the gap acting as an electrode, either as derived from the analytical thin layer cell (TLC) model or from QBIOL simulations. Note that QBIOL simulations with the two-walls acting as electrodes, i.e., forming an electrochemical nanogap, are also available.

The TLC model⁶⁴ is developed in details in section S9, but briefly, it predicts the shape of CVs depending on the confinement length $z_{gap}$, the diffusion coefficient $D$, the MH electron transfer rates and the voltage sweep rate $v$. In particular, it can compute the dimensionless potential shift $\xi = \frac{q}{k_B T} (E_p - E^0)$ (with $q$ the elementary charge, $k_B$ the Boltzmann constant, $T$ the temperature, $E^0$ the standard electrochemical potential and $E_p$ the potential of the oxidation peak) for every ($\mu$, $\mu\Lambda$) coordinate couple, with $\mu = z_{gap} \sqrt{\frac{qV}{k_B T D}}$ and $\mu\Lambda = k_s z_{gap} / D$ ($k_s$ being here the relevant electron transfer rate), that can be thought as dimensionless sweep rate and electron transfer rate, respectively (details section S9).

As shown in Fig. 2, QBIOL is in very good agreement with the model (see also Fig. S22) except for $\log(\mu) < 10^{-1}$ and $\log(\mu\Lambda) < 10^{-2.5}$, at the bottom left of Fig. 2. This corresponds to very small sweep rates combined with very slow electron transfer rates, in which cases very few events occur at

the electrode, exposing the limit of electron counting in QBIOL when simply too few events occur over the simulated times to reproduce the macroscopically expected values.

Since the position and rates applied to the electroactive particle are known at each moment, QBIOL is able to reproduce concentration profiles. If simple cases, such as a free particle confined between two electrodes are now trivial$^{57,65}$, QBIOL offers a unique insight for complex systems, such as end-attached Fc DNA strands (Fig. S4).

**Chronoamperometry.** Chronoamperometry techniques are widely used to obtain information on the rate of electron transfer, with applications in biology, biosensors and electrochemistry. QBIOL implements chronoamperometry as described in section S7.2, with an example of chronoamperometry on a redox-labeled tethered DNA double strand T20 Fig. 3a. Plaxco et al. made experiments on tethered DNA strands of various lengths and showed an oscillating pattern¹⁴, attributed to the DNA helix rotating the electrochemical head periodically closer to the electrode (Fig. 3b). Chronoamperometry experiments on QBIOL were carried out on double strand DNA from 8 to 50 base pairs. The apparent rate ($k_{app}$) was obtained as the decay rate on the current versus time curves (Fig. 3a). We observe the same periodicity than in ref. 14 (≈11 base pair) for $k_{app}$ with respect to the number of base pair (Fig. 3b, c).

The apparent decay trend of $k_{app}$ on the QBIOL curve, though absent on the experimental data, was previously observed$^{43,44,66}$ and, simply explained, attributed to shorter strands spending more time close to the electrode. In other words, $k_{app} \propto \rho \propto 1/N_{bp}$, with $N_{bp}$ the length of the DNA strand in number of bases. Considering the error bars and the difficulty of the experiment, we predict that reiterating the experiment designed by ref. 14 with more intermediate bases and statistical reproductions should render the decay visible experimentally.

<figure><img src="image_6.png" /><figcaption>**Fig. 2 | QBIOL-generated CV oxidation peak's voltage heatmap.** This heatmap generated from QBIOL cyclic voltammogram (CV) data represents the dimensionless oxidation peak's voltage $\xi$ as a function of the dimensionless parameters $\mu\Lambda$ ($\propto k_s$) and $\mu$ ($\propto v$) (other maps for current peak and current peak/sweep rate</figcaption></figure>

available in section S8). The variations of $\mu$ and $\mu\Lambda$ control the behavior of the CVs, with the crossed coordinates marking the shown CVs. The black dashes are obtained from the TLC analysis discussed section S9. ($z_{gap}$ = 60 nm).
<caption>**Fig. 3** | E-DNA chronoamperometry: comparison between QBIOL and experiment.</caption>

a Chronoamperometry of a dt20ds DNA. (E-E⁰): – 50 → 0 mV, with a single exponential decay fit.
b Schematics showing the rate change to be expected from the rotation of the DNA helix with the number of base pair. c Measurement of kapp for double-stranded DNA between Nbp = 8 and 50 base pair. A periodic pattern is visible in both experimental (reproduced from [¹⁴]) and QBIOL data. On the latter, the additional decay observed is due to shorter strands spending more time close to the electrode. Simulation parameters: MH rates, $\rho H^2/h = 4 \times 10^{-6}$ eV, z = 11.5 Å.

<figure><img src="image_7.png" /><figcaption>**Fig. 4** | Square-wave voltammetry: comparison between QBIOL and experiment. a Square wave voltammetry (SWV) voltage profile, with an enlarged version showing SWV parameters. Inset: schematics of the Fc-DNA-alkyl molecule tethered on a gold electrode. b Experimental SWV and CV (after baselines removal) compared to QBIOL numerical experiment with the same parameters. The CV currents are scaled ten times for an easier comparison. Inset: raw SWV and CV data. SWV parameters: A = 20 mV, f = 100 Hz, v = 0.1 V/s. CV parameters: v = 0.1 V/s. Simulation parameters: MH rates, $\rho H^2/h = 4 \times 10^{-6}$ eV, z = 4.2 Å.</figcaption></figure>

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We observe a difference in phase that can be explained by a difference of linker, by the location of the electrochemical tag on the DNA head and by the type of electrochemical tag used (methylene blue in the experiment, Fc in the simulation). The difference in amplitude of kapp is due to the use of a Fc molecule in the simulation, with a single-step electron transfer. Indeed, a full study to assess reorganization energy for the methylene blue used in the original article, which involves the exchange of a proton, remains to be done. The methylene blue shows an apparent electron transfer rate orders of magnitude lower than Fc⁶⁰ explaining the difference here observed. This also shows that QBIOL is a powerful tool to explore and optimize the ideal number of bases for a given E-DNA sensor.

**SWV.** Square wave voltammetry (SWV) (Fig. 4a) has been widely used for its ability to resolve a signal where CVs were too noisy or impeded by parasitic capacitive currents. However, the complex nature of the current answer to SWV waveforms prevents in practice the quantitative interpretation of experimental results, in spite of existing models⁶⁷⁻⁶⁹. As a result, empirical calibrations are typically used¹³,⁷⁰,⁷¹, using CVs as a base to take advantage of the increased sensitivity of SWV. QBIOL provides here a simulation platform to estimate both CVs and SWVs from complex systems, such as tethered electrochemical probes¹⁶,⁷⁰.

As an application example, we study a case where one wants to calibrate the SWV signal with CV measurements. We have performed experiments on a self-assembled monolayer (SAM) of dT35 double stranded DNA, as illustrated on Fig. 4a, diluted with hexanethiol alkyl chains (details in SI). Figure 4b shows the currents recorded for both the CV and the SWV for such a SAM, where the amplitude of the CV peak is roughly ten times smaller than that of the SWV. QBIOL simulations of both experiments with the same parameters are also shown on Fig. 4b, with N = 2.76 × 10¹⁰ molecules (determined from the CV) and z = 4.2 Å for the $e^{-\beta z}$ factor in the

electrochemical rate expression (with β the tunneling decay coefficient), coherent with the length of the hexanethiol used here. QBIOL shows a 1/10 ratio between CV and SWV peaks, similar to the experimental data, and also yields very close currents compared to the experimental values. These results show that QBIOL can be used to realistically simulate SWV and obtain quantitative information from SWV measurements.

**AC voltammetry.** Alternative current (AC) voltammetry (AC voltammetry) superimposes a linear sweep with a sinusoidal wave to study the frequency response of the interrogated electrochemical system. From this one can also address the frequency isolation techniques⁷²,⁷³ as well as impedance spectroscopy⁷⁴⁻⁷⁶, where the excitation frequency is swept. It allows for to discrimination of charge transfer rates of different origins and separates the different contributions to the system's impedance, with various applications in biology, batteries, fuel cells, corrosion, nanopores and porous materials. Most notably, high-harmonics are of particular interest as they almost completely remove the contribution of the double layer⁷³. AC voltammetry is available in QBIOL as illustrated Fig. 5, where we simulate a Fc free particle in a 60 nm gap, reproducing the increase of readability for high harmonics with the increase of the excitation amplitude, as shown experimentally here⁷³.

Recent developments emphasize the importance of AC voltammetry techniques, as they allow to recover superior information compared to DC techniques, whether it is through advanced analysis of frequency-isolated currents⁷³ or, as discussed further in this paper, by high-frequency capacitive amplification⁷⁷.

**Electrochemical shot-noise.** Experimental evidences were recently brought for the measurement of electrochemical shot-noise on a single-electrode device taking advantage of the stability and homogeneity of
<figure><img src="image_9.png" /><figcaption>**Fig. 5** | AC voltammetry QBIOL simulations. AC voltammetry with *f* = 70 Hz, *A* = 40, 80 or 160 mV, *v* = 0.1 V/s, *zgap* = 60 nm. a Applied voltage (zoom in inset as well as schematics of the experiment). b Current versus time. c Fourier transform of</figcaption></figure>

the current, clearly showing well defined high harmonics for higher excitation amplitudes. d Fourth harmonic envelope versus voltage for different excitation amplitudes.

electrochemical SAMs⁵⁵. As QBIOL inherently accounts for stochastic processes, it is particularly well-suited to simulate such noise experiments. The reproduction of this experiment in QBIOL, done by measuring the current over time at different voltages, is shown Fig. 6, showing the low-frequency noise versus voltage. The data in QBIOL are obtained using a prefactor on the rates to account for the undecanethiol spacer, corresponding to an equivalent distance between Fc and the surface of the electrode of ≈9.1 Å, in reasonable agreement with literature⁷⁸. The experimentally measured noise baseline is not generated in QBIOL and added from its measured experimental values to QBIOL's noise results for comparison with the experiment.

The results are in good agreement with experimental data reproduced from ref. 55. Interestingly the discrepancies on the width at half maximum between the model proposed in the original article and experimental data, attributed to interactions with neighboring molecules and electronic coupling variations, is here absent. This tends to indicate that a refining of the analytical model for electrochemical molecules is required, which we plan to address in a dedicated work.

**Designing biosensors.** If previous experiments showed archetypal DNA sequences, QBIOL is able to simulate any sequence such as bio- logically relevant aptamer sequences and can be used as an engineering tool. E-DNA probes consist of an end-attached electrochemical marker on a DNA strand or aptamer strand and have various biosensing

<figure><img src="image_10.png" /><figcaption>**Fig. 6** | Voltnoisogram: comparison between QBIOL and experiment. Comparison between experimentally recorded PSD (at 10 Hz) and QBIOL at low frequency. Inset: schematic of the experiment. Simulation parameters: BV rates, *dt*sample = 3.3e-5 s, *k*₀ = 4 × 10⁻⁷ s⁻¹, *z*gap = 1 Å, prefactor = 1.85e-4, *N* = 7.5 × 10¹⁰ molecules, noise background = 4 × 10⁻²⁶ A²/Hz. Experiment: ferrocene undecanethiol SAM diluted with undecanethiol alkyl chains with in total ≈7.5 × 10¹⁰ Fc molecules addressed in 0.5 M NaCO₄ on microelectrodes (0.78 mm²). Original data and detailed protocol in ref. 55.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7** | QBIOL as a tool for biosensors' design. a Total hydrogen bond energy of SYL3C for different $z_{gap}$. Inset: peak height at the energies indicated by arrows versus $z_{gap}$. b Probability of presence of the Fc head within the gap. The height of images are to scale with the indicated gap. Scale bars: 2 nm. Simulation parameters: MD</figcaption></figure>

simulated time = 20 µs, salt = 0.5 M, T = 20 °C. c Corresponding QBIOL CVs with the following parameters: $v = 1$ V/s, 10 mV per step, MH rates, $\rho H^2 = 4 \times 10^{-6}$ J, $z = 1$ nm in $e^{-R_c}$. Insets: top left, voltage and current peak shift with $z_{gap}$, bottom right, a schematics of the simulated experiment.

applications¹²⁻¹⁶. They have been used for the past decade as incredibly versatile biosensors, for food, metal, environment or cancer marker detection⁷⁹⁻⁸¹.

We show here a case study for the design of such a biosensor based on a SYL3C aptamer (SYL3C sequence. 5'-fc-cac tac aga ggt tgc gtc tgt ccc acg ttg tca tgg ggg gtt ggc ctg-(ch₂)₃-sh-3') for the EpCAM marker on the cancer cell membrane¹².

The aptamers are grafted on a gold electrode with nano pillars to suspend the cell, leaving the aptamer at a proper distance to both reach the surface of the cell and this electrode. The detection is made by monitoring the electrochemical current amplitude during cyclic voltammograms, as it would decrease when the aptamer matches the EpCAM receptor on the cell membrane and thus spend less time near the surface, decreasing the net current at the electrode. A crucial parameter in the development of the sensor was: what should be the height of the nano pillars to ensure that the aptamer can move freely without being compressed against the electrode, while still reaching the surface of the cell? If this was solved experimentally by trial and error, QBIOL now offers a tool to probe this question ahead of such experiments.

Figure 7 shows simulations of a single confined SYL3C strand with different $z_{gap}$. The total hydrogen energy bond Fig. 7a, important indication for the hairpins configurations of aptamers, and the probability of presence Fig. 7b suggest that for a confinement down to 5 nm, the detection loop¹²,⁸² for EpCAM sensing is dominant most of the time. CVs are also simulated for the corresponding gaps, Fig. 7c. There is a clear transition from 4 nm downward visible on hydrogen bonds, probability of presence and CVs, indicating a change in conformation. CVs remain the same from 10 nm gap, where the molecule is hardly confined, down to 5 nm where the peak clearly starts shifting toward a surface-only CV. This is due to higher probability of presence close to the electrode for more confined molecules.

It agrees with experimental findings where a 5 nm-high optimum gap was estimated between the surface of the cells and the electrode where SYL3C were grafted¹². The corresponding CVs Fig. 7c show a shift of the current peak, due to different probabilities of presence at the interface for different confinements. The presence of polyethylene glycol oligomer that hinder the access to the surface in the real experiment is not included in QBIOL at this stage, but practically results in an impossibility to reach the electrode closer than $z \approx 1$ nm, which is translated into a reduction of the electron transfer rates by a factor $e^{R_c} \approx 22,000$. The simulation of a dense layer of laterally interacting Fc-DNA chains on an electrode is also possible (Fig. S3), but since the Fc aptamer layer was loosely packed here, it was not especially relevant for this particular example. However, it can be an important feature for biosensor engineering.

If for this specific DNA sequence, the probability of presence near the interface plays the main role over the final current response, it is not true in general: for a given DNA sequence, the timescale of opening/closing of hairpins is for example not known in advance, and electron transfer rates can be of similar magnitudes. It is especially true for more complex wave-forms applied to the electrode. Note that any DNA sequence can be used for simulation within QBIOL. Furthermore, QBIOL could be extended to simulate proteins by implementing a protein MD library, such as LAMMPS⁸³.

## Pushing electrochemistry at the limit

Electrochemistry at the nanoscale is often hindered by noise, and QBIOL is a natural tool to investigate such limits. A global perspective was initiated by Gao et al.⁸⁴ across a wide variety of electrochemical systems ranging from nanopores to nanoscale electrochemical imaging, including neurotransmitter release and single particle electrochemistry. Figure 8a represents
<figure><img src="image_12.png" /><figcaption>**Fig. 8** | Electrochemical current limits. Minimum faradaic current resolved $\Delta i$ versus its measurement bandwidth $\Delta t^{-1}$ for different electrochemical systems (adapted from ref. 84). See Tables S5 to S17 for numerical values and methods.</figcaption></figure>

the original data gathered by Gao et al., as well as QBIOL voltammetry simulations and other works investigated by the authors (details listed section S10). The original study suggested a measurement limit for electrochemical currents set from a shot noise definition based on the variation of the number of measured molecules changing charge state *N* over a bandwidth $\Delta t^{-1}$ and on the “Limit of Quantification”$^{85}$ recommending a ten times bigger measurement than the background noise. It translates practically in a limit current $\Delta i = 100\sigma_p$, with $\sigma_i = \sigma_N q/\Delta t$ (black dashes on Fig. 8), $\sigma_i$ the current shot noise (blue line on Fig. 8) and $\sigma_N$ the variation of the number of elementary charges measured during $\Delta t$. If this definition of shot noise is debatable given the range of experiments considered, QBIOL simulations and literature data suggest that this limit is not absolute. In fact, many experiments were possibly constrained by their current amplifier (red small-dash line, Fig. 8) rather than true electrochemical noise.

In redox-cycling experiments, the electrochemical current and the background current (as compared in ref. 85) differ in nature. The electrochemical current results from redox cycles down to a single molecule$^{86}$, while the background current likely reflects the limit of the experiment's current amplifier. In contrast, voltnoisograms (Fig. 6) analyze a large number of molecules to determine their shared rate properties$^{25}$. A smaller number of molecules would proportionally reduce the current noise, highlighting the often-overlooked role of molecular count in measurements.

In addition to the considerations on the number of molecules, QBIOL simulations, free of any background noise, point toward lower current limits for CVs and voltnoisograms (blue spheres on Fig. 8). Experimentally, scanning microwave microscopy (SMM) (SMM, blue circles on Fig. 8) also demonstrates that it is possible to break through the aforementioned limit thanks to its capacitive amplification and frequency isolation scheme$^{77}$. QBIOL is also able to reproduce the SMM addressing scheme (see section S7.4) and highlights the potential improvements for SMM (red circle on Fig. 8), although the large difference in bandwidth is due to the absence of parasitic capacitances in QBIOL, experimentally ubiquitous and still challenging to avoid. Overall, this highlights the importance of alternative amplification schemes, such as those employed in redox cycling and SMM, as well as the potential of frequency-isolation techniques to reach lower current limits, often synonymous with a much smaller number of molecules, thus giving access to the stochastic properties of the system. Such advancements would benefit applications like single-entity measurements$^{87}$ or DNA synthesis, where precise, localized measurements are critical, for example, in DNA data storage technologies$^{17,18}$.

## Toward quantum transport devices

We have so far focused on (bio-)electrochemical applications, but the architecture of QBIOL makes it suitable for any system where stochastic electron transfer occurs at the nanoscale. As an example, we chose to

reproduce the following experiment carried out on a single-electron transistor. Ubbelohde et al. managed to measure precisely the electron statistics in a single-electron transistor (represented Fig. 9a), using a quantum point contact to monitor the presence or absence of electron within the QD of the transistor$^{88}$. They manage not only to measure accurately the current, but also the higher-order moments of noise with precision and in agreement with analytical models. By using the rates reported in their work in QBIOL and adjusting time constants to match their experiment, we can reproduce remarkably well the experimentally measured electron distribution statistics, including the second moment (Fig. 9b) and the third moment (Fig. 9c) in the frequency domain. This shows that, provided the proper electron transfer rates, QBIOL is very versatile and can be structurally very easily adapted to quantum devices of interest for solid-states and molecular organic electronic communities.

## Web interface

A web interface of QBIOL is available at https://qbiol.org. Up to date implemented features cover most of the examples shown in this work for anyone to test. In the future, QBIOL is meant to address most bioelectrochemical systems, in particular bioengineering applications, covering for example stochastic diffusion of particles in DNA synthesis$^{18}$ or transistors aptasensors$^{15}$.

## Discussion

QBIOL is able to tackle the challenge of electrochemical current simulations on large timescales thanks to its hybrid approach using coarse-grained MD and stochastic electron transfer simulations. QBIOL simulations are conducted in two stages. First, MD libraries simulate the system over timescales representative of the target current measurement. Second, electron transfer probabilities are calculated over time using this spatial data, looping over it as necessary (see section S5.3). It effectively decouples electron transfer from structural conformation changes, providing a key simplification for computation speed. While it currently cannot model intermolecular electron transfers and shares some limitations with MD for systems with significant spatial evolution over long timescales, this approach allows for reaching very long timescales compared to MD alone, with a very good accuracy on the studied systems.

The electron transfer rates used in QBIOL reflect the state of the art in the biomolecular electron transfer rates understanding$^{44–46}$. Thanks to its flexible structure, QBIOL is able to take into account future refinements or user-input custom rates.

While MD-derived presence probabilities might seem sufficient to estimate current characteristics in simple cases, the full QBIOL approach is mandatory when molecular motion, conformation changes or electron transfers cover overlapping ranges. This can typically be the case for
<figure><img src="image_13.png" /><figcaption>**Fig. 9** | Solid-state electron counting: comparison between QBIOL and experiment. Reproduction from QBIOL simulations of the results obtained by Ubbelohde et al.⁸⁸. a Schematic representation of the experiment, where the state of the QPC allows to recover independently the electrons flowing in both leads. b Spectral noise and cross-correlation of the currents taken from ref. 88 and c the equivalent obtained from QBIOL, with analytical models derived in the original article in dashed lines. The average dwell time $1/(\Gamma_L + \Gamma_R)$ is well recovered (grayed zone) in QBIOL. d Third</figcaption></figure>

order fano factor (frequency dependent skewness) taken from ref. 88 and e the equivalent obtained from QBIOL. In both cases the bottom part is obtained from the analytical model and the top part is from experimental data in (d) and QBIOL data in (e). (QBIOL simulation parameters: $\Gamma_L$ = 13.23 kHz, $\Gamma_R$ = 4.81 kHz, binning window = 40 µs, $dt_{MD}$ = 2 µs, $dt_{sample}$ = 2.77 s × 1000 threads ≈ 47 min simulated.) Original figures reproduced from ref. 88 (licensed under CC BY-NC-SA 3.0).

aptamers with hairpins forming at different time scales. Furthermore, a purely probabilistic approach would fail to capture the statistical distribution of transfer events, which provides critical insights beyond mean current values.

The QBIOL simulation time currently needed to reproduce Fig. 3 is about 4 days, including the MD and the electrochemistry. While this will be improved, it already advantageously compares to weeks of experimental work and therefore, shows that QBIOL can already be used as a screening technique.

## Conclusion

QBIOL is the first numerical tool, accessible to all, aiming to generate any (bio)electrochemical experiment that can be directly compared to real experiments. Starting from stochastic processes (molecular and charge transfer) at its core while addressing computational challenges arising from the spread in timescales between molecular motion (in the order of picoseconds) and electrical measurements (in the order of seconds), QBIOL proves to be the way to provide state-of-the-art quantitative results. It has already contributed to unveiling the operational mechanism of E-DNA sensors⁴⁴, the ballistic Brownian motion of DNA under nanoconfinement⁴⁵ and the suppression of the reorganization energy for tethered DNA via the probability of presence at the metal interface⁴⁶. It allows direct comparison to a large variety of experiments, enabling anyone to unveil new science, engineer biosensors or simply for educational purposes.

We believe that with the continuous progress of molecular dynamic simulations, AI tools for macromolecular recognition and the computing power of GPUs; such numerical experiments will surpass the number of real bioelectrochemical measurements in the next decade. It will also be

transposed to other biosensing or imaging techniques, becoming a key tool for today and tomorrow's innovations, such as transistor aptasensors¹⁵ or large-arrays of nanoelectrodes for parallel E-DNA synthesis¹⁸.

## Methods

Extensive description on how QBIOL works is given in the supplementary information available for this manuscript. A brief depiction of how QBIOL works is given hereafter. QBIOL uses existing MD libraries to obtain the position over time of a redox marker. The simulation is run long enough so that the obtained trajectory of the redox marker over time can be considered representative of the system under study. For example, for a freely diffusing particle in a box, the simulated time *t* is so that $t \gg \frac{z^2}{k_B T}$. The electron transfer rate is calculated beforehand for a subset of voltages and positions, both for Butler-Volmer and for MH formalism. A simple interpolation in space is done to get the electron transfer rates at the simulated positions and potentials. We use rejection sampling to speed up the estimation of actual electron transfers based on the relative transfer rates. From the number of electrons transferred over a given simulated duration, the current at the electrode can be derived.

## Data availability

Datasets are available upon request to the authors and through the following repository: https://doi.org/10.6084/m9 figshare.29382152.

## Code availability

QBIOL simulations are accessible through a publicly available web interface at https://qbiol.org. This access allows anyone to reproduce the results
reported in this work. The underlying source code is not publicly released at this time, but the polymer analysis code is available upon reasonable request.

Received: 27 March 2025; Accepted: 3 July 2025;

Published online: 19 July 2025

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Acknowledgements

The authors would like to thank Pierre Fauret and Christophe Zanon,
engineers in the LAAS laboratory, for their major contribution in the making of
the QBIOL web interface. The authors also thank the CNRS prematuration
program QBIOL, EU-Attract Unicorn-Dx, MITI Pressure and ANR ESHOT for
their financial support.

Author contributions

S.G. designed the main computational framework, conducted the
experiments and data analysis, and co-wrote the manuscript. I.M. designed
the main computational framework and developed the underlying mathe-
matical approach to the physical problem. A.D. and H.S. developed the
solid-state electron counting code and carried out the related experiments
and data analysis. M.K. implemented the DNA SAM computational frame-
work. A.F., S.H.K. and A.C. strengthened the electrochemical and physical
modeling aspects. C.D. performed the formal electrochemical thin-cell
analysis and co-wrote the manuscript. N.C. conceived the original idea,
contributed to the computational framework, and co-wrote the manuscript.
All authors engaged in active discussions, reviewed the results, and pro-
vided critical feedback on the final version of the manuscript.

**Competing interests**

The authors declare no competing interests.

**Additional information**

**Supplementary information** The online version contains
supplementary material available at
https://doi.org/10.1038/s42004-025-01603-1.

**Correspondence** and requests for materials should be addressed to
Simon Grall or Nicolas Clément.

**Peer review information** *Communications Chemistry* thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

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