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REVIEW 4 major objections 4 minor 15 references

Microtubules: dynamics, soliton waves, some roles in the cell

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read One microtubule equation yields all three known soliton types

desk verdict A clear but derivative review of the author's own soliton work; all key derivations are self-cites and the claimed numerical support is nowhere to be seen. read the letter →

arxiv 2501.10614 v1 pith:IK2DCNFQ submitted 2025-01-18 physics.bio-ph

Slobodan Zdravković This is my paper · ORCID
classification physics.bio-ph
keywords microtubulessolitonsnonlineardynamicsu-modelkinkbreathersbell-typeferroelectricW-potential
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

This paper argues that the nonlinear dynamics of a microtubule can be captured by a single simplified Hamiltonian, the u-model, and that its equation of motion supports all three known kinds of solitonic waves: kinks, bell-type solitons, and breathers. The kinks and the bell-type wave come from the continuum approximation, while the breather comes from a semi-discrete approximation that reduces the dynamics to a nonlinear Schrödinger equation. The paper claims these solitons are plausible candidates for information carriers along microtubules, especially in stable neuronal microtubules. If that is right, the mechanical and electrical degrees of freedom of tubulin dimers provide a concrete biophysical basis for cellular information processing.

What carries the argument

The load-bearing object is the u-model Hamiltonian of Eq. (1), a one-dimensional ferroelectric chain of electric-dipole dimers with nearest-neighbor coupling and an on-site W-potential. The term $(A/4)u_n^4-(B/2)u_n^2+C u_n$ gives the combined potential two minima, corresponding to two possible dimer orientations, and provides the nonlinearity that generates the soliton solutions. Depending on the mathematical approximation applied to the resulting equation of motion, the same Hamiltonian produces either the continuum ODE (6), whose solutions are kinks and a bell-type soliton, or the nonlinear Schrödinger equation (5), whose localized envelope solution is a breather.

What would settle it

Measure the equilibrium orientation angles between tubulin dimers and the protofilament axis in a real microtubule. The model's combined W-potential has two minima, so it predicts two distinct preferred angles, while the simplified geometry in the paper shows all dimers aligned with the protofilament. Observing only one orientation, or angles far from the model's prediction, would show that the potential generating the kinks, bell-type soliton, and breather is not the right physical description.

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Extended reading notes

Core claim

The paper's central claim is that all three known kinds of solitonic waves appear as solutions of the same discrete dynamical equation for microtubules. Starting from a Hamiltonian in which each tubulin dimer is an electric dipole with one effective degree of freedom along a protofilament, the equation of motion (2) is solved in two approximations. The continuum approximation gives three kink/antikink solutions describing a localized transition between two dimer orientations, plus a bell-type soliton that appears only when viscosity is neglected. The semi-discrete approximation gives a localized modulated wave, the breather, whose envelope is about 200 nm wide and covers about 25 dimers. The paper states that all these analytical results are numerically supported and concludes that the solitons are candidate information carriers along microtubules.

Load-bearing premise

Everything rests on the assumption that a real microtubule behaves like a single chain of electric-dipole units with only one vibrational direction each, interacting only with immediate neighbors in a potential that has not been measured; if that picture is wrong, the predicted waves are artifacts of the math.

Editorial extensions

If this is right

  • Kink solutions represent a moving transition between two dimer orientations, so the model implies that a localized bit-like domain wall can travel along a protofilament.
  • The bell-type soliton exists only when the viscosity coefficient is set to zero, predicting that low-damping conditions are needed for this kind of signal to propagate.
  • The breather has a concrete spatial scale, about 200 nm across 25 dimers, which gives a testable size for any localized excitation observed in microtubules.
  • If these solitons carry information, stable neuronal microtubules could plausibly support processing and storage of biological information, as the paper suggests.
  • The two-minima W-potential predicts two measurable equilibrium angles between dimers and the protofilament direction.

Reading between the lines

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

  • The paper's model applies to a single protofilament, so a natural extension the author does not pursue is how solitons on neighboring protofilaments couple through the electric field that appears in Eq. (1).
  • Because the model parameters A and B are left to be estimated, the predicted breather width or kink transition interval could be fitted to future experimental observations to determine those constants.
  • The two dimer orientations implied by the W-potential suggest that an applied electric field could switch a dimer between orientations; this switching idea is implicit but not developed in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper reviews the structure and biological roles of microtubules (MTs) and presents the "u-model" Hamiltonian for a single protofilament (Eq. (1)), from which a discrete equation of motion (Eq. (2)) and its continuum (Eq. (3)) and semi-discrete (Eq. (5)) approximations are asserted. The author then exhibits three soliton solutions of these equations: kink solitons (Fig. 6), a bell-type soliton (Fig. 7), and a breather (Fig. 8), and concludes that "all the three known kinds of solitonic waves have been found in MTs" and that these solitons are "possible candidates for information carriers along MTs".

Significance. If the central claim were established, the paper would provide a compact review supporting the hypothesis that nonlinear excitations in microtubules can serve as information carriers. The manuscript clearly presents the biological background and introduces the relevant physical quantities of tubulin dimers. However, the key equations are imported from earlier work without derivation, the model parameters are never connected to biophysical values, and the asserted numerical support is not shown. As a result, the significance of the paper for real microtubules is not demonstrated; at most it shows soliton solutions of an abstract Hamiltonian.

major comments (4)
  1. [Eq. (1), Nonlinear dynamics of MTs] The model parameters A, B, k, Q, E, and gamma are never assigned numerical values or estimated from experimental data. The text explicitly states that A and B 'are parameters that should be determined or, at least, estimated.' Without such estimates, the solutions of Eq. (3) and Eq. (6) cannot be connected to a real protofilament, and the statement that the three solitons have been 'found in MTs' is an overclaim.
  2. [Figs. 6 and 7, Solitonic waves in microtubules] The kink solutions are plotted for rho=2, sigma=0.31, while the bell-type soliton is obtained only for rho=0, sigma=0.34 or 0.1. Since rho is proportional to the viscosity coefficient, the three soliton types are not shown to coexist in a single physical regime. The physical meaning of the chosen dimensionless parameters and the experimental conditions they represent are not explained.
  3. [Eq. (7) and final paragraph, Solitonic waves in microtubules] The breather solution in Eq. (7) has all its parameters deferred to Ref. [9], and the derivation of Eq. (5) is also delegated to prior references, including Ref. [14] by the author. This means the central result is not self-contained and cannot be independently verified from the present manuscript. Moreover, the claim that 'All these analytical results have been numerically supported' is unsupported: no numerical data, methods, or code are presented, and the figures show only analytical curves.
  4. [Abstract and Conclusion] The abstract and conclusion state that all three soliton kinds 'have been found in MTs' and are 'possible candidates for information carriers.' Given that the model parameters are not tied to biophysical measurements and the numerical evidence is absent, these conclusions go beyond what the manuscript establishes. The final paragraph itself concedes that experiments would be needed to 'prove or disapprove the theoretical expectation regarding W-potential,' which undercuts the strong phrasing used earlier.
minor comments (4)
  1. [Introduction, Fig. 1 caption] The caption reads 'A tubulin dimers, a protofilament and a microtubule' and should be corrected to 'A tubulin dimer, a protofilament, and a microtubule.'
  2. [References, Ref. [4]] Reference [4] is an Internet search URL rather than a proper citation to a specific source; the origin of the figures should be documented more reliably.
  3. [Abstract] The phrase 'Three different kinds of them are known in the moment' should be revised to 'at present' or 'currently.'
  4. [Solitonic waves in microtubules, historical paragraph] The name 'John Scott Russel' is a misspelling; the correct spelling is 'John Scott Russell.'

Circularity Check

2 steps flagged · score 8.0 of 10

Central 'solitons in MTs' result is imported from the author's own prior papers: NLSE (5), parameter values for (6), and the breather (7) are deferred to Refs. [9] and [14], with numerical support asserted, not shown.

  1. self citation load bearing [Solutions of Eq. (2), paragraph introducing NLSE Eq. (5)]
    "The complete procedure and important explanations can be found in Ref. [14]. ... In this particular case they turned out to be constants [9]. Hence, we only need the equation for the function F(ξ) and this is a well-known solvable nonlinear Schrödinger equation (NLSE) (5), where the dispersion coefficient P and the coefficient of nonlinearity Q are explained in the aforementioned references."

    The transition from the discrete Eq. (2) to the NLSE Eq. (5) is the load-bearing step for the breather: it determines the envelope F whose sech form becomes Eq. (7). The paper does not carry out this transition; it delegates the 'complete procedure' to Ref. [14] and the closed forms of F0 and F2 to Ref. [9], both by the present author or co-authors. Thus the breather 'found in MTs' is not derived or checked in this paper; it reduces to accepting the author's own prior derivation of the same equation for the same model.

  2. self citation load bearing [Solitonic waves in microtubules, Eqs. (6)-(7) and Figs. 6-8]
    "The values of the parameters α, ρ and σ can be found in aforementioned references. ... A final step is derivation of Eq. (5). Even though this is PDE its solution exists [13,14]. ... This interesting solution is ... (7), where the expressions for all the parameters can be found in Ref. [9]."

    The kinks and bell-type solitons plotted in Figs. 6-7 are solutions of Eq. (6) with parameter values (ρ=2, σ=0.31; ρ=0, σ=0.34/0.1) whose definitions are not given but 'can be found in aforementioned references'; the breather Eq. (7) explicitly sends all parameter expressions to Ref. [9]. Refs. [9] and [14] are the same author's prior works. The conclusion 'all the three known kinds of solitonic waves have been found in MTs' therefore restates those self-cited results unchanged, with no numerical data or artifacts shown to support the sentence 'All these analytical results have been numerically supported.'

full rationale

This paper does not actually derive its central result. The discrete equation (2) is transformed into the NLSE (5) by citing Ref. [14] and Ref. [9]; the ODE (6) is solved with parameter values 'found in aforementioned references'; and the breather (7) has all its parameter expressions 'found in Ref. [9]'. Refs. [9] and [14] are the present author's own prior works (JNMP 2011 and Appl. Math. Comput. 2016), so the claim that kinks, bell-type solitons, and breathers 'have been found in MTs' is a report of self-cited results, not a self-contained derivation or an independently verified numerical demonstration. The sentence 'All these analytical results have been numerically supported' is asserted without figures, data, or code, so it cannot serve as an independent check. Additionally, the model parameters A and B in the W-potential are explicitly said to 'be determined or, at least, estimated' and the plotted solutions use arbitrary dimensionless values (ρ=2, σ=0.31; ρ=0, σ=0.34/0.1), so the 'found in MTs' claim is not tied to biophysical parameter values. These latter points are correctness/evidential gaps rather than circularity; the circularity score is driven by the self-citation chain that carries Eqs. (5)-(7) and the conclusion. The paper is not circular in the sense of a fit renamed as a prediction; no fitting occurs. It is circular in the weaker, load-bearing sense: the derivation chain that is supposed to support the conclusion is deferred to the same author's earlier papers, and the present text adds no independent verification. Score: 8, because the central result is effectively imported from the author's own prior derivation chain without new support; 10 would require the equations to be identical to the inputs by definition, which is not quite the case.

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

The model depends on several assumptions and unmeasured parameters inherited from prior work; the paper introduces no new entities. The central soliton solutions rest on the ferroelectric single-PF Hamiltonian and on approximations whose validity is not tested here.

free parameters (6)
  • A (W-potential coefficient) = not given
    Appears in Hamiltonian Eq. (1); the text says A and B are parameters that should be determined or at least estimated, from Refs [8,9].
  • B (W-potential coefficient) = not given
    Same paragraph as A; central to the shape of the potential in Eq. (1).
  • gamma (viscosity coefficient) = not given
    Introduced through F_v = -gamma u_dot in Eq. (3); appears in parameter rho in Eq. (6), and the bell-type soliton exists only for rho=0, i.e., zero viscosity.
  • E (internal electric field) = not given
    Appears in C = Q E in Hamiltonian Eq. (1); no value or measurement cited in this paper.
  • Q (excess charge of dimer) = not given
    Appears in C = Q E in Eq. (1); taken as positive, no value given.
  • k (intra-dimer stiffness parameter) = not given
    Appears in the harmonic coupling term in Eq. (1); no value provided.
assumptions (7)
  • domain assumption MTs can be regarded as ferroelectric
    The W-potential terms in Eq. (1) are introduced because the MT is assumed ferroelectric; no experimental confirmation cited here.
  • domain assumption A single protofilament Hamiltonian suffices, with neighboring protofilaments represented only by a background electric field
    The text states longitudinal contacts appear much stronger than inter-PF contacts, so the model is 'practically for a single PF only'.
  • domain assumption Nearest-neighbor approximation for dimer interactions
    The Hamiltonian in Eq. (1) includes only n and n+1 coupling; the text calls this 'very common in physics'.
  • domain assumption One degree of freedom per dimer
    The u-model projects dimer motion onto the protofilament direction; the text says all existing models assume one degree of freedom, though attempts with two are in progress.
  • standard math Continuum approximation u_n(t) -> u(x,t) with Taylor expansion
    Used to obtain Eq. (3); validity depends on long-wavelength modes.
  • domain assumption Small oscillations and modulated-wave ansatz Eq. (4)
    The semi-discrete derivation assumes epsilon << 1 and separates a continuous envelope from a discrete carrier; leads to NLSE Eq. (5).
  • domain assumption Solitons are candidates for information carriers
    Stated in the conclusion as a prediction; no experimental or numerical basis is provided in this paper.

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Cite this review

Pith. "Pith review of Microtubules: dynamics, soliton waves, some roles in the cell." pith.science (2026). https://pith.science/paper/IK2DCNFQ

@misc{pith2026250110614,
  author       = {Pith},
  title        = {Pith review of: Microtubules: dynamics, soliton waves, some roles in the cell},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IK2DCNFQ}},
  note         = {Machine review of arXiv:2501.10614}
}
read the original abstract

In the present paper we deal with nonlinear dynamics of microtubules (MTs). The structure and role of MTs in cells are explained. One model explaining MT dynamics is explained. Solutions of the crucial nonlinear differential equation depend on used mathematical procedures. Two of them, continuum and semi-discrete approximations, are explained. Finally, these solutions are shown and discussed. They are solitonic waves. Three different kinds of them are known in the moment. They are kink solitons, breathers and bell-type solitons.

Figures

Figures reproduced from arXiv: 2501.10614 by the authors.

Figure 1
Figure 1. A tubulin dimers, a protofilament and a microtubule [4]. MTs are the major part of cytoskeleton ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. A nucleus and MT network in a eukaryotic cell [4] [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Motor proteins move along MTs carrying cargo [4] [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (2 more)
Figure 7
Figure 7. Figure 7: A bell-type soliton for   0.34 (blue) and   0.1 (red). As a conclusion we can say that all the three known kinds of solitonic waves have been found in MTs. All these analytical results have been numerically supported [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: A localized modulated soliton (breather) CONCLUSION AND FUTURE RESEARCH The three different solitons have been predicted as possible candidates for information carriers along MTs. They are shown in Figs. 6-8. A crucial question is their stability, which is very importa…

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Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [9]

    Zdravkovi ć, S

    S. Zdravkovi ć, S . Zeković, A. N. Bugay , M. V. Satari ć, Applied Mathematics and Computation, 2016, 285, 248–259

    work page 2016

  2. [14]

    Zdravković, Journal of Nonlinear Mathematical Physics, 2011, 18 (Suppl

    S. Zdravković, Journal of Nonlinear Mathematical Physics, 2011, 18 (Suppl. 2), 463–484

    work page 2011

  3. [1]

    Hameroff, R

    S. Hameroff, R. Penrose, Physics of Life Reviews, 2014, 11, 39–78

    work page 2014

  4. [2]

    S. Sahu, S. Gh osh, K. Hirata, D. Fujita, A. Bandyopadhyay, Applied Physics Letters, 2013, 102, 123701

    work page 2013

  5. [3]

    Havelka, M

    D. Havelka, M. Cifra, O. Kučera, J. Pokorný, J. Vrba, Journal of Theoretical Biology, 2011, 286, 31–40

    work page 2011

  6. [4]

    (https://www.google.rs/search?q=microtubule&biw=1680&bih=913 &tbm=isch&tbo=u&source=univ&sa=X&sqi=2&ved=0ahUKEwi_4d uGgdfMAhXlDsAKHWVACEAQsAQIJw)

    Taken from Internet on the 13th of May, 2016. (https://www.google.rs/search?q=microtubule&biw=1680&bih=913 &tbm=isch&tbo=u&source=univ&sa=X&sqi=2&ved=0ahUKEwi_4d uGgdfMAhXlDsAKHWVACEAQsAQIJw)

    work page 2016

  7. [5]

    Cifra, J

    M. Cifra, J. Pokorny, D. Havelka, O. Kučera, BioSystems, 2010, 100, (2010) 122–131

    work page 2010

  8. [6]

    Cassimeris, N

    L. Cassimeris, N. K. Pryer, E. D. Salmon, Journal of Cell Biology, 1988, 107, 2223-2231

    work page 1988

Show all 15 references
  1. [7]

    S. R. Hameroff, R. C. Watt, Journal of Theoretical Biology, 1982, 98, 549-561

    1982

  2. [8]

    M. V. Satarić, J . A.Tuszyński, R. B. Žakula, Physical Review E, 1993, 48, 589-597

    1993

  3. [10]

    Zdravković, M

    S. Zdravković, M. V. Satarić, S. Zeković, Europhysics Letters, 2013, 102, 38002

    2013

  4. [11]

    Zdravković, M

    S. Zdravković, M. V. Satarić, A. Maluckov, A. Balaž, Applied Mathematics and Computation, 2014, 237, 227-237

    2014

  5. [12]

    Remoissenet, Physical Review B, 1986, 33, 2386-2392

    M. Remoissenet, Physical Review B, 1986, 33, 2386-2392

    1986

  6. [13]

    Dauxois, M

    T. Dauxois, M. Peyrard, Physics of Solitons, Chapter 3, Cambridge University Press, Cambridge, UK, 2006

    2006

  7. [15]

    Zdravković, G

    S. Zdravković, G. Gligorić, Chaos 2016, 26, 063101

    2016

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