A Landauer Formula for Bioelectronic Applications

D\'avid P. Jelenfi; Eszter Papp; G\'abor Vattay; M\'at\'e T. Veszeli

REVIEW 4 major objections 4 minor 29 references

A Landauer Formula for Bioelectronic Applications

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Protein-junction conductance is set by vibrations and Boltzmann weights, not by tunneling through the molecule.

desk verdict A genuinely new strong-decoherence conductance formula that would be important if its unverified relaxation-rate assumption holds; the experimental claims outrun the evidence. read the letter →

arxiv 1909.01196 v1 pith:EWJTZD3Q submitted 2019-09-03 cond-mat.dis-nn cond-mat.mes-hallphysics.bio-phq-bio.BMquant-ph

Eszter Papp , Dávid P. Jelenfi , Máté T. Veszeli , Gábor Vattay This is my paper · ORCID

classification cond-mat.dis-nncond-mat.mes-hallphysics.bio-phq-bio.BMquant-ph

keywords LandauerformulaconductanceofbiomoleculesmetalliccontactsdecoherenceRedfieldequationelectrontransferproteinstylizedfacts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that the puzzling experimental behavior of protein junctions—conductance in the nano-Siemens range, almost no drop over 1–10 nm, and near constancy between 30 and 300 K—follows from a single generalized Landauer formula once strong coupling to vibrations is included. In the regime the authors call strong decoherence, the conductance is governed not by tunneling through the whole molecule but by the equilibrium Boltzmann occupation of molecular orbitals together with the electrode coupling strengths $\Gamma_n$. The result reduces the problem from a full quantum-chemical transport calculation to a handful of macroscopic parameters, and it recovers the familiar Landauer and weak-coupling electron-transfer limits when decoherence is absent or when contacts are weak. A sympathetic reader would care because it offers a parameter-light explanation of why long protein wires can conduct as well as they do.

What carries the argument

The engine of the derivation is the Bloch–Redfield Liouvillian $L_{nmkl}$, whose inverse feeds the generalized current formula (28) through terms like $L^{-1}_{kknn}$. In the strong-decoherence limit the paper evaluates that inverse by perturbation theory: the zero eigenvalue of the isolated molecule's Redfield operator is shifted by the contacts to a small value, and the inverse is dominated by the reciprocal of that shift, $$$L^{{-1}}$_{nnmm} \approx -\frac{\hbar\,\$varrho^{0}$_{nn}}{\sum_p \Gamma_p \$varrho^{0}$_{pp}},$$ with $\varrho^0$ the Boltzmann density matrix. This single identity is what removes the microscopic detail: it turns the many couplings of the molecule to its vibrations into the statement that the electron or hole wanders through the Boltzmann-weighted orbitals and exits through whichever electrode offers the larger escape rate.

What would settle it

Compute the eigenvalues of the isolated-molecule Redfield operator $L_0$ from a realistic electron–phonon model of a protein junction. If any nonzero eigenvalue is of the same order as or smaller than $\Gamma_n$, the strong-decoherence formula (66) is not justified. Experimentally, a direct contradiction would be a symmetric two-strong-contact junction whose conductance decays exponentially with electrode separation or follows an Arrhenius factor with half the HOMO–LUMO gap.

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Extended reading notes

Core claim

The paper's central claim is that in the strong-decoherence regime, the conductance of a protein junction is given by Eq. (66), $$G = \frac{$e^{2}$}{h}\left(T_h^R(E_F)P_h^L(T) + T_h^L(E_F)P_h^R(T) + T_e^R(E_F)P_e^L(T) + T_e^L(E_F)P_e^R(T)\right),$$ where each $T$ is a tunneling term built from the orbital energies and level broadenings $\Gamma_n$, and each $P$ is an equilibrium Boltzmann probability that a charge leaves through a given electrode. The formula is derived from the Redfield equation by inverting its Liouvillian in the limit where vibrational relaxation dominates the electrode couplings, and the derivation shows that all microscopic details of the electron–vibration interaction cancel out. The remaining input is just the molecular orbital spectrum and the two contact strengths. From this the paper obtains conductance values in the $0.01$–$10$ nS range, almost no temperature dependence below the nearest orbital gap, and no systematic dependence on the distance between two strong contacts.

Load-bearing premise

The load-bearing assumption is that the vibrational relaxation rates inside the protein are all much larger than the electrode coupling strengths $\Gamma_n$, so the inverse Redfield operator is dominated by a single perturbed zero eigenvalue; the paper does not compute or estimate those relaxation rates in actual proteins.

Editorial extensions

If this is right

With two strong contacts, conductance through a single protein should stay flat as electrode separation grows from 1 to 10 nm, because none of the terms in (66) depends systematically on distance.
With one strong and one weak contact, distance dependence can reappear with an exponent set by where the HOMO and LUMO sit relative to the strong contact, matching the power-law relation observed in peptide nucleic acids.
The electron-transfer rate and the conductance are not proportional in the strong-decoherence regime: the transfer rate keeps an Arrhenius factor and decays with length, while conductance does not.
The temperature dependence, when present, should be set by the smallest HOMO/HOMO-1 or LUMO/LUMO+1 gap, not by half the HOMO-LUMO gap, which is how the Myoglobin and Cytochrome C datasets are reproduced.
Predictions for new protein junction experiments can be made from orbital energies and contact couplings computed semiempirically, avoiding expensive full quantum transport calculations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: if the strong-decoherence formula is right, isotopic substitution or matrix changes that alter vibrational relaxation rates without changing orbital energies should barely affect the conductance, since the vibration details cancel; this is a testable extension the paper does not perform.
Inference: the formula effectively describes the junction as a classical random walk on Boltzmann-weighted orbitals with electrode escape as the absorbing step, so the same logic could connect this result to environment-assisted transport models used in photosynthetic energy transfer.
Inference: the assumption $\Gamma_n \ll |R|$ could be checked directly by computing the full Redfield spectrum of a realistic electron–phonon model of a protein; real proteins may sit in the strong-decoherence regime or in a crossover where Eq. (66) needs correction terms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a real attempt at a new result, and Eq. (66) is worth taking seriously, but the paper overclaims on the experimental side and the key decoherence assumption is unchecked. I would send it to review, but the referee should push hard on the spectral gap problem.

The genuinely new thing is the strong-decoherence limit of a Redfield-generalized Landauer formula. For a closed system the Redfield Liouvillian has a zero eigenvalue corresponding to Boltzmann equilibrium; the paper perturbs that eigenvalue in the lead couplings and gets a conductance that depends only on Boltzmann weights and the Γ's, with vibrational details dropping out. That is a surprising and potentially important result, and it recovers the right coherent limit when |R|≪Γ. This is not circular: the formula is derived from stated physical assumptions, not from the data. They also provide code, so the curve in Fig. 3 is reproducible.

The soft spots are real. The perturbation step in Appendix B keeps only the shifted zero eigenvalue and assumes all other eigenmodes of L0 are fast. The paper never estimates those relaxation rates for a protein, and the parameter range used later (Γ≈0.1 eV) makes the required hierarchy Γ≪|R| far from obvious. There is also an internal tension: the Redfield tensor is itself a weak-coupling object, so "strong decoherence" has to come out of a Born-Markov term that is assumed large. Without a spectral-gap estimate or a check against a solvable model, Eq. (66) is conditional.

The experimental sections do not fix that. Temperature curves are reconstructed from two endpoints and fitted with I0/IT; the distance-exponent argument selects x=0.17 to match the measured ratio; the conductance distribution uses Gaussian couplings with means and variances set by visual optimization. Those are consistency checks, not validation. The authors mostly say that too, but the abstract and discussion lean on the experimental agreement.

Who should read it: people working on decoherence in molecular junctions and protein transport. They will find the strong-decoherence formula a useful starting point even if the quantitative claims are not yet solid. I would not cite (66) as established, but I would cite the paper as the origin of the formula.

Recommendation: send to peer review, with the requirement that the authors estimate or bound the spectral gap of the Redfield operator in a realistic model and present the experimental fits as exploratory.

Referee Report

4 major / 4 minor

Summary. The paper derives a generalized Landauer-Büttiker formula for molecular conduction starting from a Markovian Redfield master equation. After obtaining a general expression for conductance in terms of the inverse of the Redfield operator (Eq. (28)), the authors consider two limits: weak contacts, where conductance is proportional to the electron-transfer rate (recovering Nitzan-type relations), and strong contacts. The central result is the strong-decoherence limit, Eq. (66), in which the conductance depends only on the equilibrium Boltzmann distribution and the electrode coupling strengths, leading to nS-scale conductance, weak temperature dependence, and distance independence when both contacts are strong. The paper applies this formula to Myoglobin and Cytochrome C temperature data, to the PNA distance-exponent relation, and to the conductance distribution of Streptavidin.

Significance. If correct, Eq. (66) is a significant simplification: it replaces a many-parameter microscopic description with a handful of macroscopic parameters and offers a concrete mechanism (vibrational mixing) for the experimentally observed high, temperature-insensitive, distance-insensitive conductance of proteins. The derivation from the Redfield equation is transparent, and the appendix sum rules (Eqs. (80) and (98)) are useful. The paper also provides falsifiable predictions, such as distance independence for two strong contacts, and states that simulation code is provided as supplementary material. However, the validity of the strong-decoherence reduction is not quantitatively established, and the experimental comparisons are mostly consistency checks with fitted parameters, so the current evidence for the central claim is incomplete.

major comments (4)

[Appendix B, Eq. (87); Section 5.2] The approximation (87) for L^{-1} retains only the perturbatively shifted zero eigenvalue of L0, which is valid only if all other eigenvalues of L0 (the vibrational relaxation rates) are large compared with the lead couplings Γ_n. The paper never estimates these eigenvalues for proteins; Section 5.5 uses Γ in the 0.1 eV range, so the assumed hierarchy Γ ≪ |R| is a postulate rather than an established condition. This is load-bearing because Eq. (66), and the claimed distance and temperature independence, follow only in this strong-decoherence limit. The authors should provide a quantitative estimate of |R| from electron-phonon coupling parameters or a numerical check of the spectral gap in a representative model.
[Section 6.2, Eq. (75)] The explanation of the PNA exponent β_G ≈ 0.66 β_ET is obtained by choosing x ≈ 0.17 after the fact; Eq. (75) with a fitted x that matches the measured exponent is a post-hoc fit, not a prediction of the theory. Moreover, the assumption that couplings scale as Γ_HOMO/LUMO ∝ e^{-β_ET x l} introduces the electron-transfer decay exponent β_ET as an input. The section should be reframed as a consistency check or as a demonstration that a plausible x can account for the data, with x treated as a free fit parameter.
[Section 6.1, Tables 1 and 3] The fits in Figs. 1 and 2 use two adjustable parameters per curve (I0 and IT), and in the electrostatic Cytochrome C case the activation energy is taken from the experimental value 0.105 eV rather than from the computed orbital structure; the computed HOMO-LUMO gap/2 = 0.161 eV is not used. The agreement is therefore weaker evidence than the text suggests, and the activation-energy identification is partly circular. The authors should report goodness-of-fit, parameter uncertainties, and ideally make predictions for at least one curve with parameters fixed by the theory.
[Section 5.2] There is an internal tension between using the Redfield equation, which is derived under a Born-Markov weak system-bath coupling assumption, and the strong-decoherence regime Γ ≪ |R|, which requires large R. The paper should discuss the parameter regime in which both assumptions hold, or demonstrate with a specific model that the Redfield tensor can produce relaxation rates exceeding Γ without leaving its domain of validity.

minor comments (4)

[Abstract and Keywords] The phrase 'Landauer fromula' in the keywords is a typo for 'Landauer formula'.
[Section 5.3] The text contains 'LOMO' in place of 'LUMO' in the sentence about the smallest orbital energy differences.
[Section 8] 'Matlab and Phyton' should be 'Matlab and Python'.
[Figures 1 and 2] The authors state that start and end points of curves were visually extracted from original figures; they should indicate the estimated digitization uncertainty and provide the raw extracted data points in the supplementary material.

Circularity Check

2 steps flagged · score 4.0 of 10

Central strong-decoherence formula (66) is derived from the stated master-equation assumptions and is not circular; the post-hoc fits in Sections 6.2-6.3 (x≈0.17 and optimized couplings) are the only circular elements.

fitted input called prediction [Section 6.2, Distance dependence of electron transfer and conductance, around Eqs. (74)-(75)]
"Assuming x≈ 0.17 can explain the relation of exponents observed in the experiment. This suggests that the HOMO and LUMO are closer to the metal contact, which is due to the Ferrocene redox center attached to the end of the molecule in contact with the metal."

The measured exponent relation β_G≈0.66 β_ET is converted into x≈0.17 through Eq. (75), G∼e^{-β_ET(1−2x)l}, because 1−2(0.17)=0.66. The parameter x is not determined from the PNA structure or from an independent first-principles input; it is chosen so that the predicted exponent equals the observed one. Thus the claimed explanation of the exponent relation reduces to fitting the free location parameter to the datum it is supposed to explain.
fitted input called prediction [Section 6.3, Distribution of conductance, after Fig. 3]
"These concrete values are the results of optimization carried out by visually comparing the result of the calculation with the experimentally obtained data."

The coupling strengths of the STM tip and substrate (V = 0.4 eV and 0.07 eV, with variances) are optimized to the measured conductance distribution. The subsequent statement that the simulation's average, variance and shape are compatible with the experimental distribution is therefore an in-sample fit, not an out-of-sample test of Eq. (66). The agreement is built into the chosen parameters rather than being predicted from the theory alone.

full rationale

The derivation chain from Eq. (12) through Eq. (28) and from Eq. (55) through Eq. (66) is self-contained: it starts from a Redfield equation, defines the inverse Liouville operator, uses a probability-conservation sum rule, and in the strong-decoherence limit takes the perturbatively shifted zero eigenvalue to dominate L^{-1}. The resulting conductance formula depends only on Boltzmann occupations and lead couplings; no experimental stylized fact is fed into the algebra, and no load-bearing self-citation or imported uniqueness theorem is used. The unverified Γ≪|R| hierarchy and the weak-coupling status of Redfield theory are validity concerns, not circularity. The circular elements are confined to the application section: the distance-exponent explanation sets x≈0.17 so that Eq. (75) reproduces β_G≈0.66β_ET, and the conductance-distribution simulation optimizes the coupling parameters to the target data before declaring compatibility. Both are post-hoc fits rather than independent predictions, and they do not retroactively invalidate the central derivation. A full quantitative test would require independent estimates of x and of the coupling parameters, or a fit to one dataset followed by prediction on another. Score 4 reflects partial circularity in application-level 'explanations' while the core formula remains independently derived.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The theory introduces no new particles or forces. Its conceptual load is carried by the Markovian Redfield model, the Boltzmann approximation, and the strong-decoherence perturbation assumption; quantitative comparisons require fitted contact parameters (I0, IT, coupling strengths, x) in the application sections.

free parameters (7)

I0 (Myoglobin curves) = 1.0e-7, 1.0e-8, 2.0e-6 A/cm2
Fit to current density vs temperature data from Ref. [25]; Table 1.
IT (Myoglobin curves) = 3.2e-5, 6.6e-6, 2.7e-5 A/cm2
Fit to temperature-dependent part of current; Table 1.
I0 (Cytochrome C curves) = 3.7e-6, 1.1e-7 A/cm2
Fit to current data from Ref. [26]; Table 3.
IT (Cytochrome C curves) = 5.3e-6, 2.1e-4 A/cm2
Fit to temperature-dependent part; Table 3.
Tip coupling mean and std = 0.4 eV, 0.35 eV
Chosen by visual optimization to match the Streptavidin conductance distribution; Section 6.3.
Substrate coupling mean and std = 0.07 eV, 0.035 eV
Chosen by visual optimization to match the Streptavidin conductance distribution; Section 6.3.
x (fractional position of HOMO/LUMO orbitals) = 0.17
Assumed to reproduce the observed ratio beta_G/beta_ET in PNA experiments; Section 6.2, Eq. (75).

assumptions (6)

domain assumption Markovian Redfield equation accurately describes electron-vibration dynamics in the molecular junction
The derivation of the generalized Landauer formula is based on the Markovian Redfield master equation (Eq. 12), assumed in Section 2 as a practical approximation.
domain assumption Boltzmann distribution replaces Fermi-Dirac for occupied and unoccupied molecular levels
Invoked in Section 2 (Eq. 18) using the large HOMO-LUMO gap approximation; requires |E_F - epsilon_n| >> kT for relevant states.
domain assumption The molecule remains charge neutral with equal electron and hole Boltzmann partition sums
Equation (17) and the definition of Z(T) in Section 2 require equal sums over hole and electron states when the Fermi energy lies between HOMO and LUMO.
domain assumption In the strong decoherence limit, the inverse Redfield operator is dominated by the perturbed zero eigenvalue, with all other eigenvalues large compared to lead couplings Gamma_n
Appendix B derives L^{-1} in first-order perturbation theory around the equilibrium zero mode; requires Gamma_n much smaller than all relaxation rates, a condition not independently verified for proteins.
domain assumption Localized molecular orbitals couple significantly to only one electrode at a time
Used in Section 5.4 to drop cross terms and conclude distance independence; requires negligible orbital overlap between distant sites.
domain assumption Semiempirical Extended Hückel (YaEHMOP) gives reliable orbital energies near the HOMO-LUMO gap
Used in Section 8 to compute the orbital energies used for activation energy predictions in Section 6.1.

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Cite this review

Pith. "Pith review of A Landauer Formula for Bioelectronic Applications." pith.science (2026). https://pith.science/paper/EWJTZD3Q

@misc{pith2026190901196,
  author       = {Pith},
  title        = {Pith review of: A Landauer Formula for Bioelectronic Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EWJTZD3Q}},
  note         = {Machine review of arXiv:1909.01196}
}

read the original abstract

Recent electronic transport experiments using metallic contacts attached to proteins identified some 'stylized facts' which contradict conventional wisdom that increasing either the spatial distance between the electrodes or the temperature suppresses conductance exponentially. These include nearly temperature independent conductance over the protein in the 30-300K range, distance independent conductance within a single-protein in the 1-10 nm range and an anomalously large conductance in the 0.1-10 nS range. In this paper we develop a generalization of the low temperature Landauer formula which can account for the joint effects of tunneling and decoherence and can explain these new experimental findings. We use novel approximations which greatly simplify the mathematical treatment and allow us to calculate the conductance in terms of a handful macroscopic parameters instead of the myriads of microscopic parameters describing the details of an atomic level quantum chemical computation. The new approach makes it possible to get predictions for the outcomes of new experiments without relying solely on high performance computing and can distinguish important and unimportant details of the protein structures from the point of view of transport properties.

Figures

Figures reproduced from arXiv: 1909.01196 by the authors.

**Figure 1.** c. of the original paper and we show them in Fig 2. We used again the fit (73) and the parameters are in Table 3. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Current density curves of experiment Ref[ [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: Comparison of measurement data from Figure 4A of Ref.[ [PITH_FULL_IMAGE:figures/full_fig_p014_3.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Protein bioelectronics: a review of what we do and do not know

Christopher D Bostick, Sabyasachi Mukhopadhyay, Israel Pecht, Mordechai Sheves, David Cahen, and David Lederman. Protein bioelectronics: a review of what we do and do not know. Reports on Progress in Physics , 81(2):026601, 2018

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[2] [2]

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Juan M Artés, Ismael Díez-Pérez, and Pau Gorostiza. Transistor-like behavior of single metalloprotein junctions. Nano letters, 12(6):2679–2684, 2011

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Observation of giant conductance ﬂuctuations in a protein

Bintian Zhang, Weisi Song, Pei Pang, Yanan Zhao, Peiming Zhang, István Csabai, Gábor Vattay, and Stuart Lindsay. Observation of giant conductance ﬂuctuations in a protein. Nano futures, 1(3):035002, 2017

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Bintian Zhang, Weisi Song, Pei Pang, Huafang Lai, Qiang Chen, Peiming Zhang, and Stuart Lindsay. Role of contacts in long-range protein conductance. Proceedings of the National Academy of Sciences, 116(13):5886–5891, 2019

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Fabrication of bioﬁlm in nanoscale consisting of cytochrome f/2-maa bilayer on au surface for bioelectronic devices by self-assembly technique

Si-Youl Yoo, Taek Lee, Yong-Ho Chung, Junhong Min, and Jeong-Woo Choi. Fabrication of bioﬁlm in nanoscale consisting of cytochrome f/2-maa bilayer on au surface for bioelectronic devices by self-assembly technique. Journal of nanoscience and nanotechnology , 11(8):7069–7072, 2011. 17 A PREPRINT - SEPTEMBER 4, 2019

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Conductance switching in the photoswitchable protein dronpa

Katalin V Korpany, Pinky Langat, Dong Myeong Kim, Neil Edelman, Daniel R Cooper, Jay Nadeau, and Amy Szuchmacher Blum. Conductance switching in the photoswitchable protein dronpa. Journal of the American Chemical Society, 134(39):16119–16122, 2012

work page 2012