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A Simple Voltage-Modulated Markov Chain Model for the Piezo1 Ion Channel to Investigate Electromechanical Pacing

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A four-state Markov chain model of the Piezo1 stretch-activated channel quantitatively reproduces voltage-clamp inactivation and desensitization data and, in a ventricular myocyte model, qualitatively explains rate-dependent loss of…

desk verdict Sensible model idea, honest limitations, but the fitted parameter table makes the Markov chain invalid as written; re-fit needed before the pacing claims can stand. read the letter →

arxiv 2501.19366 v2 pith:KPQGOT6O submitted 2025-01-31 q-bio.QM q-bio.SC

Dennis Ogiermann , Abdulaziz Mohamed , Luigi E. Perotti , Daniel Balzani This is my paper · ORCID
classification q-bio.QMq-bio.SC
keywords Piezo1stretch-activatedionchannelcontinuous-timeMarkovchainvoltage-dependentinactivationdesensitizationelectromechanicalpacingventricularmyocytemodelmechano-electricfeedback
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

What the paper tries to establish is that the voltage dependence of the Piezo1 stretch-activated channel is a primary mechanism behind the loss of mechanical capture in electromechanical pacing. To do this, the authors build a four-state continuous-time Markov chain whose transition rates depend exponentially on both pressure and transmembrane voltage, fit it to patch-clamp recordings, and show it reproduces voltage-dependent inactivation, weak rectification, desensitization during pressure trains, and the reset of desensitization at positive driving forces. When the channel is added to a rabbit ventricular myocyte model, the simulated cell shows the same qualitative pattern seen in Langendorff-perfused rabbit hearts: repeated mechanical stimuli depolarize until capture is lost, with the number of captured stimuli depending on pacing rate and on interleaved electrical stimuli. The authors are explicit that the integrated model does not quantitatively reproduce all pacing protocols with a single parameter set, and they conclude that Piezo1 alone may not be sufficient, but that the channel's voltage modulation is likely a major contributor. A reader should care because mechanical pacing is used in CPR and emergency bradycardia treatment, and no mechanistic explanation previously existed for why it works and why it fails.

What carries the argument

The engine of the paper is a four-state continuous-time Markov chain with states $O$ (open), $C$ (closed), $I_1$ (fast-inactivated), and $I_2$ (slow-inactivated). Transition rates have the Eyring-like form $r_i \exp(c_i^m p + c_i^e \Delta\mu)$, where $p$ is pressure and $\Delta\mu$ is the electrochemical driving force, set equal to the membrane voltage because the Piezo1 reversal potential is near zero. Pressure dependence is kept on the closed-to-$I_1$ and $I_1$-to-open transitions, and the $I_2$-to-open transition depends on the product $p\,\Delta\mu$; every edge also carries a voltage term, and detailed balance is enforced through parameter constraints. The channel's open probability drives separate Ohmic Na$^+$, K$^+$, and Ca$^{2+}$ currents that are added to the Mahajan-Shiferaw myocyte model, which is what lets voltage and pressure jointly shape capture and loss of capture.

What would settle it

A decisive check is to compute the generator matrix entries from Table 1 at physiological pressure and voltage; if any off-diagonal rate is negative, the channel is not a Markov chain and the fitted equations as written cannot be integrated as probability dynamics.

Watch

Extended reading notes

Core claim

The central claim is that Piezo1's electromechanical response can be captured by a four-state continuous-time Markov chain in which every transition rate depends exponentially on pressure and on the electrochemical driving force, here identified with the transmembrane voltage. On that basis the paper reports quantitative agreement with the voltage-clamp recordings of Moroni et al. for voltage-dependent inactivation and weak rectification, and qualitative agreement for desensitization during repeated pressure stimuli and for the reset of desensitization when pressure is applied at positive driving forces. When the channel is coupled to the Mahajan-Shiferaw rabbit ventricular myocyte model, the simulations reproduce the essential pacing phenomenon: mechanical capture is lost after a number of mechanical stimuli, and that number depends on pacing rate and on alternating electrical stimuli, matching the direction of the Quinn-Kohl observations. The authors also state plainly that no tested parameter combination reproduces every experimental pacing outcome quantitatively, and that Piezo1 alone may not suffice.

Load-bearing premise

The model is a probability model only if all transition rates are non-negative for every pressure and voltage used, but the fitted parameters include negative rate coefficients and the paper gives no rule for interpreting a negative rate.

Editorial extensions

If this is right

  • The voltage-modulated Markov chain gives a mechanistic explanation for why positive electrochemical driving forces reset Piezo1 desensitization: at positive voltages the channel population re-enters states that can open again on the next pressure step.
  • In the Mahajan-Shiferaw cell, adding Piezo1 raises intracellular Na$^+$ and Ca$^{2+}$ by roughly 8--10%, shortens the action potential, and depresses its peak; stronger pressure steps increase Na$^+$ more than Ca$^{2+}$, a prediction that could be tested ion-selectively.
  • The integrated model reproduces the qualitative ranking that alternating mechanical and electrical stimuli (2:1 E:M) causes faster loss of capture than 3:1 E:M for some parameterizations, supporting the claim that voltage modulation contributes to capture loss.
  • Because no single parameter set matches all Quinn-Kohl protocols quantitatively, the authors conclude that mechanisms beyond Piezo1, possibly other stretch-activated currents or tissue-level conduction, are needed for full quantitative agreement.

Reading between the lines

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

  • If the transition rates from Table 1 are evaluated literally at physiological pressures and voltages, some generator-matrix entries are negative; the model would not define a Markov chain, and the pacing results may depend on how the solver handles these entries. Refitting with non-negativity constraints is the natural next step.
  • The paper's finding that a lower reversal potential of roughly $-30$ mV would be needed for reset during passive filling suggests that the assumption $\Delta\mu = $ membrane voltage is a sensitive point; if recent estimates near $-15$ mV hold, the model's reset dynamics could change substantially.
  • A direct experimental falsification would be to measure Piezo1 open probability under a two-pulse protocol at a series of positive voltages and compare the reset time constant to the model's prediction; because the model was fit only to current traces, the state probabilities themselves are an unvalidated prediction.
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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

3 major / 4 minor

Summary. The paper proposes a four-state continuous-time Markov chain model of the Piezo1 ion channel with voltage- and pressure-dependent transition rates, fits the rate parameters to voltage-clamp current traces from Moroni et al. (2018), and integrates the resulting channel model into the Mahajan-Shiferaw rabbit ventricular myocyte model. The integrated model is then used to simulate idealized cardiac cycles and electromechanical pacing protocols modeled on the experiments of Quinn and Kohl (2016). The authors report that the channel model reproduces a wide range of Piezo1 experimental observations, and that the cell-level model qualitatively captures some rate-dependent features of mechanical capture and loss of capture, although it does not reproduce all experimental findings.

Significance. If the model were valid, it would be a valuable contribution: it is, to the authors' knowledge, the first Markov-chain-based Piezo1 model integrated into a ventricular myocyte model, it is provided in open CellML form with reproducible simulation scripts, and it is calibrated against independent patch-clamp data rather than only against the target pacing experiments. The experimental comparisons in Figures 3 and 4 show qualitatively reasonable behavior, and the authors are transparent about the model's limitations. However, the paper's central quantitative claim is undermined by a fundamental mathematical flaw in the fitted rate parameters, as detailed in the major comments. The pacing-level conclusions are also weakened by the fact that the two cell-level free parameters (scaling and prNaK) are fitted to the very Quinn-Kohl capture counts the model is then said to explain.

major comments (3)
  1. [§2.1.2, Table 1] The fitted rate coefficients r4, r5, r6, r7, and r8 in Table 1 are negative, while the transition rates are defined as k_i = r_i * exp(...) with exponential factors that are always positive. For example, k4 = r4 exp(cm4 p + ce4 Δμ) is negative for all p and Δμ because r4 = -0.008945307. A continuous-time Markov chain requires every transition rate to be non-negative for all states and all conditions; negative off-diagonal entries in the generator mean that probability mass is not conserved and the process has no stochastic interpretation. Since all subsequent simulations (Figures 3-7) use these parameters, the central claim that the model quantitatively reproduces experimental observations is not supported as stated. This is a load-bearing error that affects the entire model, not a cosmetic issue.
  2. [§2.2 (detailed balance constraint)] The authors state that detailed balance is enforced by requiring r1*r3*r5 = r2*r4*r6, leading to r2 = r1*r3*r5/(r4*r6). With the printed parameter values, r1, r2, r3 are positive while r4, r5, r6 are negative, so the left-hand side r1*r3*r5 is negative and the right-hand side r2*r4*r6 is positive. The equality cannot hold. This means either the table is misprinted or the constraint was not actually enforced during optimization. In either case, the manuscript presents an internally inconsistent parameter set, and the relationship between the stated constraint and the reported fit is not credible. The authors need to re-examine their optimization code and either report a parameter set that satisfies detailed balance with all non-negative r_i or explain and justify the discrepancy.
  3. [§3, Figs. 6-7 and §4] The cell-level claim that the model can 'qualitatively explain some of the experimental observations from Quinn and Kohl' is weakened by the in-sample nature of the parameter choice. The free parameters scaling and prNaK are explicitly varied 'to reproduce the total number of captured mechanical stimuli recorded in the experiments' (Section 3), and Figure 7 shows the loss surface over these parameters. The paper then reports that no parameter combination reproduces all experimental findings, and specifically that no simulation reproduces the observation that alternating electrical and mechanical stimuli cause faster loss of mechanical capture than mechanical pacing alone. The abstract's phrasing ('qualitatively reproduce some aspects') is accurate but the discussion should more clearly state that the pacing-level result is a partial in-sample fit rather than a validated prediction, especially since the channel model itself is invalid for the reasons above.
minor comments (4)
  1. [§1 and §4.2] There are several typos and spacing errors, e.g., 'an thus' (Section 2.1.3), 'T able 1' (Table 1 caption), 'eletromechanical' (Section 4.2), and a run-on 'the proposedmodelisabletoreproduce' (Section 3). A careful proofread is needed.
  2. [Fig. 1] The transition rate labels in Figure 1 are hard to read: the subscripts and exponents (e.g., ce7Δμp) are not clearly formatted, and the figure does not indicate that the parameters in Table 1 are for normalized p and Δμ even though the text says p and Δμ are normalized in the optimization. Please clarify.
  3. [§2.2] The loss function in Eq. (6) is described as a 'Huber-type Lasso loss' with H(x) = min(|x|, x^2). The notation 'min(|x|, x^2)' is not standard; for |x|>1 this is x^2, which grows unboundedly, so it is not the usual Huber loss. Please define the functions precisely to avoid confusion.
  4. [§4.1] The sentence about Buonocunto et al. mentions 'their experimental work suggests that the reversal potential ... around -15 mV', but the citation [8] is a computational characterization paper; please verify whether this is an experimental or computational finding.

Circularity Check

1 steps flagged · score 4.0 of 10

Cell-level pacing agreement is obtained by fitting the two free parameters to the very capture counts reported as reproduced; the channel-level comparisons are externally calibrated and not circular.

  1. fitted input called prediction [Section 3, electromechanical pacing study (Figs. 6-7), and Section 4 (Discussion & Conclusion)]
    "First, a coarse parameter grid was utilized (not shown) to narrow down the parameter space to reproduce the total number of captured mechanical stimuli recorded in the experiments. In this first sweep of the parameter space, we identified that the conductance scaling parameter should be between 0.13 and 0.23, while the relative NaK contribution should be between 0.0 and 0.16. This region was then analyzed using a fine grid. ..."

    The free parameters scaling and prNaK are selected in a coarse-to-fine sweep whose explicit target is the Quinn-Kohl capture numbers N, and the same N values are then presented as reproduced by the model. The agreement is therefore a fit rather than an independent prediction: the parameter search was constructed to make the outcome match. The fitted values (e.g., scaling = 0.23, prNaK = 0.067) are also carried into the idealized-cycle study, so the later mechanistic claims share the fitted input. The channel-level fits to Moroni et al. and the comparison with Wu et al. are external and non-circular; only the cell-level 'reproductions' reduce to the fitting target.

full rationale

The Piezo1 Markov chain itself is fitted to voltage-clamp data from Moroni et al. and then confronted with held-out figures from the same work and with independent data from Wu et al. without re-tuning; that part is self-contained and externally falsifiable. The circularity is confined to the cell-level study: scaling and prNaK are declared unknown, then a coarse-to-fine parameter sweep is explicitly aimed at reproducing the Quinn-Kohl capture counts, after which the paper states that the model reproduces those counts and 'qualitatively explain[s]' loss of capture. This is a fitted input called a reproduction rather than a prediction. The negative fitted rates and the detailed-balance sign inconsistency in Table 1 are serious mathematical validity problems, but they are correctness concerns, not circularity in the derivation sense, so they do not raise the circularity score. Overall partial circularity (4/10) is appropriate because the central channel model retains independent empirical content despite the fitted cell-level pacing results.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central model rests on 18 fitted kinetic parameters plus two grid-searched cell-level parameters (scaling and prNaK) and several hand-set assumptions (reversal potential, pressure-stretch mapping). The most consequential unstated assumption is that the fitted negative rate coefficients still define a valid Markov chain. No new physical entities are introduced.

free parameters (5)
  • Markov chain rate and coupling parameters (r1-r8, ce1-ce8, cm2, cm4) = See Table 1; 18 values, e.g., r1=0.02634342, ce1=-2.3753126, cm2=12.101605
    Fitted to normalized current traces from Moroni et al. 2018 (mouse Piezo1 in N2a cells) using a Huber-Lasso loss with random-restart L-BFGS.
  • scaling (conductance scale, related to NPz1) = Study range 0.13-0.23; best region shown in Fig. 7
    Scales the total Piezo1 conductance in the myocyte model; varied over a grid to reproduce the number of captured mechanical stimuli in Quinn-Kohl experiments.
  • prNaK (relative Na-K permeability in EK,s) = Study range 0.0-0.16
    Adjusts the sodium-potassium Nernst potential in the Mahajan-Shiferaw model; grid-searched together with scaling against Quinn-Kohl capture counts.
  • Characteristic pressures for diastolic stretch (25 mmHg) and saturating stretch (70 mmHg) = 25 mmHg and 70 mmHg
    Chosen from a linear stretch-pressure mapping where 30% stretch equals 70 mmHg and 10% diastolic stretch equals 25 mmHg; this mapping is not derived from experiments.
  • Piezo1 reversal potential Er = 0 mV
    Assumed from the literature to identify the electrochemical driving force with the membrane voltage; the authors note in Section 4.2 that a lower reversal potential (about -30 mV) would be needed to reset the channel in the cell model.
assumptions (6)
  • domain assumption Detailed balance holds for the Piezo1 Markov chain under arbitrary electrochemical gradients.
    Adopted from dwell-time analyses [80] and enforced in Section 2.2 as a constraint on the rate parameters.
  • domain assumption The four-state Markov chain topology (O, I1, I2, C) with the given pressure- and voltage-dependent rates is an adequate representation of Piezo1 gating.
    Based on the earlier models of Bae et al. and Lewis et al. and on modal analysis of coarse-grained simulations; introduced in Section 2.1.2.
  • ad hoc to paper The rate expressions r_i exp(...) with the fitted parameters define valid continuous-time Markov chain transition rates.
    This is assumed implicitly in Section 2.1.2, but Table 1 contains negative r_i values for r4 through r8, violating the non-negativity requirement for Markov chain rates. No constraint or alternative interpretation is provided.
  • domain assumption Piezo1 reversal potential is 0 mV, so the electrochemical driving force equals the membrane voltage.
    Stated in Section 2.1.3 and Figure 1 caption, citing references [30, 16]. The authors acknowledge in Section 4.2 that this may need revision.
  • domain assumption The Piezo1 current is Ohmic and consists of independent K+, Na+, and Ca2+ components with no interactions between ions in the pore.
    Assumed in Section 2.1.3 (Eq. 1) to allow separate flux terms in the Mahajan-Shiferaw concentration equations.
  • ad hoc to paper Axial stretch of the cardiomyocyte maps linearly to the pressure variable in the channel model, with 30% stretch equal to 70 mmHg and 10% diastolic stretch equal to 25 mmHg.
    Introduced in Section 3 for the idealized cardiac cycle; this mapping is not experimentally determined and affects the predicted Piezo1 activation during diastole.

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Pith. "Pith review of A Simple Voltage-Modulated Markov Chain Model for the Piezo1 Ion Channel to Investigate Electromechanical Pacing." pith.science (2026). https://pith.science/paper/KPQGOT6O

@misc{pith2026250119366,
  author       = {Pith},
  title        = {Pith review of: A Simple Voltage-Modulated Markov Chain Model for the Piezo1 Ion Channel to Investigate Electromechanical Pacing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPQGOT6O}},
  note         = {Machine review of arXiv:2501.19366}
}
read the original abstract

Piezo1 ion channels are voltage-modulated, stretch-activated ion channels involved in a variety of important physiological and pathophysiological processes, as for example cardiovascular development and homeostasis. Since its discovery, it has been known that this type of ion channel desensitizes when exposed to stretch. However, recent experiments on Piezo1 ion channels have uncovered that their stretch response is qualitatively different when exposed to positive electrochemical driving forces, where the desensitization is reset. In this work, we propose a novel voltage-modulated mathematical model of Piezo1 based on a continuous-time Markov chain. We show that our Piezo1 model is able to quantitatively reproduce a wide range of experimental observations. Furthermore, we integrate our new ion channel model into the Mahajan-Shiferaw ventricular cardiomyocyte model to study the effect of electromechanical pacing at the cellular scale. This integrated cell model is able to qualitatively reproduce some aspects of the experimental observations regarding the rate-dependence of electromechanical pacing protocols. Our studies suggest that the Piezo1 ion channel is an important component that significantly contributes to the electromechanical coupled response of cardiomyocytes.

Figures

Figures reproduced from arXiv: 2501.19366 by the authors.

Figure 1
Figure 1. Continuous-time Markov chain of the proposed Piezo1 model extending the model of [41]. Analogously to the original work, the proposed model contains an open state O, fast and slow inactivation states I1 and I2, and a closed state C. p denotes the pressure and ∆µ the electrochemical driving force acting on the channel. Since the reversal potential of PIEZO channels is hypothesized to be approximately 0 (i.e., Er = 0,… view at source ↗
Figure 2
Figure 2. Schematic of the proposed lumped parameter cell model based on the Mahajan-Shiferaw rabbit ventricular cardiomyocyte model [49]. The new Piezo1 asso￾ciated currents (IKPz1, INaPz1 and ICaPz1) are highlighted inside boxes with bold-red fonts. JSR is the junctional sarcoplasmatic reticulum and NSR is the non-junctional sarcoplasmatic reticulum. The remaining currents and fluxes are defined as in the original model [49… view at source ↗
Figure 3
Figure 3. Simulation of the Piezo1 Markov chain model vs. experimental results from [52, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Simulation of the Piezo1 Markov chain model vs. experimental results from [52, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Idealized pacing of the ventricular cardiomyocyte model at 3 Hz using the original cell model and the cell model with the addition of the proposed Piezo 1 ion channel. The Ca concentrations in different compartments (i, d, and s refer, respec￾tively, to the intracellul…
Figure 6
Figure 6. Figure 6: Differences in the total number of captured mechanical stimuli between the computational results and the experimental observations for different choices of prNaK and scaling parameters in the proposed model (results are reported as number of cap￾tured mechanical stimul…
Figure 7
Figure 7. Figure 7: Left: Surface of the loss function constructed as the sum of the squared differences from [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.