Theory of cell size regulation during migration in adhered cells

Jonathan E. Ron; Ram M. Adar

REVIEW 3 major objections 4 minor 43 references

Theory of cell size regulation during migration in adhered cells

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that cell polarization is easier to establish over larger contact areas, so cell swelling drives faster migration once a critical contact area is crossed.

desk verdict Solid minimal framework uniting volume regulation, spreading, and polarization; the claimed critical-area prediction is currently fitted, not tested. read the letter →

arxiv 2505.16750 v3 pith:WEUCJQAA submitted 2025-05-22 physics.bio-ph

Jonathan E. Ron , Ram M. Adar This is my paper · ORCID

classification physics.bio-ph

keywords cellmigrationvolumeregulationpump-leakmodelmembranepotentialcontactareapolarizationbifurcationneutrophilswelling

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

Most models of cell migration assume size is fixed, but experiments show that cells lose volume while spreading and swell when activated, with migration speed tracking cell size. This paper builds a minimal theory coupling volume, contact area, and migration, tracing size to membrane potential and actin-driven forces. The central result is that polarization requires a spatial gradient of actin-regulatory cues, and such gradients are easier to establish over larger contact areas; hence a critical contact area exists below which cells move randomly and above which persistent migration switches on. The theory yields a velocity-versus-area curve with a maximum and reproduces the measured swelling and acceleration of activated neutrophils. If it is right, it explains why swollen neutrophils migrate faster and identifies contact area, not water content, as the quantity that controls speed.

What carries the argument

The central object is the polarization expansion $p(\tilde u \cos\phi) \approx p_1 \tilde u + p_3 \tilde u^3 + p_5 \tilde u^5$ inserted into the force balance $2\pi\Gamma u = \int_0^{2\pi} f_a(\phi)\cos\phi\,d\phi$. It converts the advection-diffusion competition for polarity cues into a bifurcation for the Péclet number $\tilde u = uR/D$. The critical radius $R_c = \Gamma D/(f_0 p_1)$ is where the factor $\Gamma D/(f_0 R)$ crosses the linear gain $p_1$, and the signs of $p_3$ and $p_5$ decide whether the onset is continuous or abrupt. Supporting machinery includes the pump-leak volume relation $V = V_d + (V_m - V_d)/(1 - e^{\psi})$, a linear-response membrane potential $\psi(t)$, a contact-angle relaxation toward the force-balance value, and a spherical-cap area-volume relation that couple cell size to the active force $f_0$.

What would settle it

Measure polarity-cue profiles around the contact ring while perturbing actin flow, and check whether the force anisotropy really has the assumed fixed-coefficient form; or, holding volume fixed while changing only contact area through substrate adhesiveness, test whether persistent migration turns on exactly at $R_c = \Gamma D/(f_0 p_1)$ and follows the predicted $u(R)$ curve.

Watch

Extended reading notes

Core claim

The paper's central claim is that cell size regulates migration through a polarization mechanism, not through volume per se. The contact area sets how easily an asymmetry in polarity-cue concentration is maintained: advection of cues by retrograde cytoplasmic flow competes with diffusion, and the Péclet number $\tilde u = uR/D$ grows with the product of speed and contact radius. Expanding the force anisotropy $p(\tilde u \cos\phi)$ as $p_1 \tilde u + p_3 \tilde u^3 + p_5 \tilde u^5$, the steady-state force balance produces a bifurcation: for $R$ below $R_c = \Gamma D/(f_0 p_1)$ the unpolarized state $u=0$ is stable, while above $R_c$ a polarized branch $u(R)$ emerges, smoothly when $p_3<0$ and discontinuously when $p_3>0$. Because $u = \tilde u D/R$, the speed first rises with area and later falls at large radii. The same framework, with volume set by pump-leak ion transport and area by a spherical-cap force balance, reproduces the observed neutrophil sequence of fast spreading with slight volume loss followed by swelling that enlarges the contact area and lifts migration speed, and it matches the finding that velocity correlates more strongly with contact area than with volume.

Load-bearing premise

The velocity-area prediction hinges on treating the polarization response as a fixed polynomial whose first two coefficients are fitted to the same velocity data they are used to explain; if those coefficients vary with cell type or state, the predicted critical area is not an independent prediction.

Editorial extensions

If this is right

Above the critical contact area, persistent migration switches on: cells below it diffuse randomly while cells above it move with a speed that initially grows with area.
Migration speed is non-monotonic in contact area, peaking at $R = 3R_c/2$ in the smooth-onset case, so sufficiently large cells are predicted to slow down again.
The sign of $p_3$ controls the transition type: $p_3<0$ gives a smooth supercritical onset, while $p_3>0$ gives an abrupt subcritical jump and bistability between resting and motile states.
Swelling accelerates neutrophils by enlarging the contact area rather than by adding water, and the experimental data confirm that velocity correlates with area more strongly than with volume.
Random motility below threshold is tied to the critical area through $\langle u_\xi^2\rangle \sim 1/A_c$, giving a measurable relation between a cell's random speed and the onset of directed motion.

Reading between the lines

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

Beyond the paper, the same mechanism should apply to other motile cell types: modulating substrate adhesion at fixed volume should shift the onset of persistent migration and rescale the velocity-area curve.
If the subcritical branch ($p_3>0$) is realized, the model predicts hysteresis, so a swollen cell would keep migrating when its area is reduced below $R_c$ until a lower threshold is reached; this can be tested in swelling-shrinking cycles.
The theory suggests a design rule for synthetic motile cells: increasing contact area lowers the polarization threshold, while raising friction $\Gamma$ or cue diffusivity $D$ raises it.
Fitting Eq. (10) to single-cell tracks during the swelling phase would turn the reported correlation into a dynamical prediction for how speed should track measured area.

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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

This is a solid theory paper with an honest core and an overreach in its framing. What's new: the authors couple pump-leak volume regulation to spreading mechanics and to a polarization-migration feedback, and show that the coupling produces a critical contact area above which persistent migration appears. The claim that swelling promotes migration by enlarging the contact area rather than by raising the volume is a distinct, testable mechanism, and it is consistent with the neutrophil data in Nagy et al. The analytic work in Appendices A-C is careful; the stability result in Appendix B (δ1>0 ensures stability) is correct and clearly explained.

The soft spot is the word 'prediction.' Eq. (9) is a phenomenological expansion of the polarization function p in odd powers of the Péclet number, with coefficients p1, p3, p5 simply posited. That is a legitimate modeling choice, but in Fig. 4, p1 and p3 are fitted to the very velocity data the model is claimed to reproduce. The critical area Ac ≈ 1.2 A(0) is therefore set by the fit, not independently predicted. The stress-test note is right: Rc = ΓD/(f0 p1) is an interpolated parameter for this dataset. The paper would be much stronger if the coefficients were tied to a microscopic advection-diffusion calculation (even in a limiting case), or if the theory were tested out-of-sample (e.g., against a data set where the velocity-area curve is not used to set p1). A smaller but real issue: in the p3 > 0 case used in the fits, the expansion requires p5 < 0, but p5 is never reported, so the branch is incompletely specified. This is a minor omission since p3 ≈ 10^-6 makes the subcritical jump tiny, but it should be fixed.

The correlation analysis in Appendix D supports the area-not-volume mechanism, but it is presented without error bars or significance testing. That is more of a minor point, since the main claim is the mechanism, not the correlation strength.

Bottom line: this is a useful proposal and deserves a serious referee. I would suggest the editor send it out, with the expectation that the authors soften the 'prediction' language, report p5, and ideally add an out-of-sample test or an independent estimate of p1 from polarity-cue measurements. I'd bring it to a reading group where the topic is cell biophysics; it will generate a good discussion on what counts as a prediction.

Referee Report

3 major / 4 minor

Summary. The paper presents a minimal theoretical framework coupling cell volume, contact area, and migration velocity in adhered cells. Volume is described by a pump-leak model with mechanosensitive and activation-induced membrane-potential changes; contact area is obtained from a spherical-cap geometry with viscoelastic tension dynamics; migration emerges from a polarization feedback expressed as a phenomenological expansion of the active-force anisotropy in the Péclet number. The theory predicts a critical contact area above which persistent migration is possible, and a velocity–area curve with a maximum. The authors compare their predictions with neutrophil spreading/swelling and migration data from Nagy et al. and report good agreement, including a correlation analysis showing that velocity correlates more strongly with contact area than with volume.

Significance. If the central claims hold, the framework would provide a useful minimal explanation for how cell-size changes, particularly contact area, control the onset and speed of persistent migration, and it would organize existing observations of neutrophil swelling and motility into a single coupled dynamical picture. The analytic derivations in Appendices A–C are internally consistent, and the stability analysis in Appendix B correctly shows that δ1 > 0 stabilizes the volume–area coupling. The correlation analysis in Appendix D is a useful check of the proposed area–velocity link. However, the predictive content is weakened by the fact that the key polarization coefficients p1 and p3 are fitted to the same experimental velocity data used to validate the velocity–area curve; as a result, the critical area is a fitted threshold rather than an independent prediction.

major comments (3)

[§III, Eq. (9), Fig. 4] The critical contact area Ac ≈ 1.2 A(0) is not an independent prediction because the coefficient p1 in Eq. (9) is fitted to the same velocity data that Eq. (10) is claimed to explain. With the reported post-activation values ΓD/f0 = 0.26 µm and p1 = 0.033, one obtains Rc = ΓD/(f0 p1) ≈ 7.9 µm and Ac = π Rc² ≈ 195 µm², which is about 1.27 A0; the stated value of 1.2 A0 is therefore set by the fit. The manuscript should either provide an independent determination of p1 (e.g., from measured polarity-cue profiles or a microscopic advection-diffusion model) or explicitly re-frame the critical-area statement as a fitted threshold rather than a prediction.
[§III, Appendix C, Eq. (C13)] The parameter set reported for Fig. 4 (p1 = 0.033, p3 = 10⁻⁶) corresponds to the p3 > 0 case, which requires the fifth-order coefficient p5 in Eq. (C13) to specify the unstable branch and the threshold R*_c via Eq. (C16). However, p5 is not reported anywhere, so the subcritical branch and the jump velocity u*_c used in the experimental comparison are incompletely specified. Please report the value of p5 used to generate the curve in Fig. 4c, along with the resulting R*_c and u*_c.
[§II, Eq. (9)] The expansion p(ũ cos φ) ≈ p1 ũ + p3 ũ³ + p5 ũ⁵ is introduced as a phenomenological step, with no derivation from the advection-diffusion feedback described in the text. Because this expansion directly determines the bifurcation threshold Rc = ΓD/(f0 p1) and the entire velocity–area curve, the sign and magnitude of p1 and p3 are load-bearing. The authors should either justify these coefficients from the underlying polarity-cue dynamics or clearly state that they are effective, cell-type-dependent parameters whose universality is untested.

minor comments (4)

[General] There are several typographical errors that should be corrected, including 'polarzied' (Abstract/Introduction), 'electrnetruality' (Section II), 'charactarized' (Appendix C), and 'concnentrations' (Appendix A).
[Fig. 3 caption] The caption reports only ΓD/f0 = 1.12 µm and p3 = ±1; the values of p1 and p5 used in the plotted curves are not given, which makes the figure difficult to reproduce.
[Appendix D] The correlation coefficients r(x,y) in Fig. 5 are presented without confidence intervals or significance tests, so the claim that velocity correlates 'significantly' more strongly with area than with volume is not quantified.
[§III] The phrase 'We consider a critical contact area of Ac ≈ 1.2 A(t = 0)' is presented as if it were a theoretical input, but the preceding text does not explain how this value arises from the model parameters; clarifying the derivation would avoid confusion.

Circularity Check

1 steps flagged · score 6.0 of 10

The critical-area and velocity-area predictions reduce to the fitted coefficients p1 and p3 of the phenomenological expansion in Eq. (9), so the central migration claim is interpolation rather than an independent prediction.

fitted input called prediction [Section II 'Cell migration and polarization', Eq. (9); Section III 'Cell migration'; Fig. 4 caption]
"Due to the many arguments that enter p (such as the protein-membrane interaction, intracellular transport, and force generation description), we phenomenologically expand it in powers of ˜u, as follows: ... ≈ p1 ˜u + p3 ˜u3 + p5 ˜u5 (9) ... Explicitly, we consider a critical contact area of Ac ≈ 1.2 A(t = 0). Once this value is achieved, u smoothly increases with the contact area, explaining the observed behavior. Fig. 4 caption: p1, p3 = 0.033, 10−6."

The critical radius is Rc = ΓD/(f0 p1) (Eq. C11), and the velocity-area curve (Eq. 10) is obtained by solving the expansion (9) for fixed p1, p3, p5. The paper does not derive p1 and p3 from the advection-diffusion mechanism; it calls the expansion phenomenological. In the comparison with Nagy et al., the parameters are set to p1 = 0.033 and p3 = 10−6 (Fig. 4 caption) so that the theoretical velocity curve matches the measured rise, and the text then 'considers' Ac ≈ 1.2 A(0) as the activation threshold. Since the threshold location and the shape of the rise are controlled by these same fitted coefficients, the agreement in Fig. 4c is an interpolation, not an independent test of the predicted critical area. For the p3 > 0 branch actually used, the curve also depends on p5 (Eqs.

full rationale

The paper's volume/spreading dynamics (Eqs. 1–6) and the area-versus-volume correlation analysis (Appendix D) are self-contained against external data and do not involve circularity. The circularity is confined to the migration section. Eq. (9) is presented as a phenomenological expansion of the polarization function p in powers of the Péclet number, with coefficients pn left free. The bifurcation threshold Rc = ΓD/(f0 p1) and the velocity-area curve Eq. (10) are then solved from this expansion (Appendix C). In the experimental comparison, p1 and p3 are chosen (Fig. 4 caption) so that the velocity curve rises where the measured velocity rises, and the text then 'considers' Ac ≈ 1.2 A(0). Thus the critical contact area—the paper's headline prediction—is not an independent, out-of-sample prediction; it is a re-parameterization of the fitted coefficients. The qualitative statement that gradients are easier to establish over larger sizes is built into the assumption p1 > 0 and the R dependence of the prefactor, not derived from a molecular model. Nevertheless, the paper does not rely on a self-citation chain or on a uniqueness theorem; the volume-regulation and spreading parts have independent content. The deficiency is a fitted-input-called-prediction in the central migration claim, warranting a partial circularity score of 6 rather than a higher score.

Assumptions & free parameters 15 free parameters · 10 assumptions · 1 invented entities

The model relies on several fitted parameters, especially the polarization coefficients p1 and p3 that set the critical area and velocity magnitude. The underlying physiology (pump-leak, mechanosensitivity) is standard, but the migration prediction is tied to a phenomenological expansion. The polarity cues are not invented ex nihilo; they reference real molecular candidates.

free parameters (15)

p1 = 0.033
First-order polarization coefficient; sets critical radius Rc = ΓD/(f0 p1).
p3 = 10^-6
Cubic polarization coefficient; determines transition type (subcritical/supercritical) and velocity magnitude.
p5 = assumed negative, not specified
Fifth-order coefficient; truncation assumption.
ψ0 = -1.53
Steady-state membrane potential at end of activation.
δ1 = 0.75
Mechanosensitivity of membrane potential to tension.
δ2 = 0.08
Activation-induced change in membrane potential.
τψ = 12 min
Membrane potential adaptation time.
τθ = 0.85 min
Spreading relaxation time.
τγ = 0.85 min
Tension relaxation time.
f0/γ0 = 1.47 (pre), 1.96 (post activation)
Active force per unit length relative to steady tension.
η/γ0 = 0.54 τγ/A0
Viscoelastic viscosity relative to tension.
ΓD/f0 = 0.34, 0.26 µm
Friction-diffusion over active force scale.
Vd = 0.3 V0
Dry volume.
Vm = 2.5 Vd
Minimal volume in isotonic conditions.
Ac = 1.2 A0
Critical contact area chosen post hoc to match onset of persistent migration in Fig. 4.

assumptions (10)

standard math Pump-leak model with electroneutrality and osmotic balance
Invoked in Appendix A to derive Eq. (1); standard physiology model.
standard math Cell shape is a spherical cap
Eq. (5) assumes the adhered cell is a spherical cap.
ad hoc to paper Linear response for potential dynamics
Eq. (2) linearizes conductance/concentration changes into a single timescale, a modeling simplification.
ad hoc to paper Maxwell viscoelastic model for tension
Eq. (4) assumes tension homeostasis with a Maxwell relaxation.
domain assumption Instantaneous channel adaptation
Assumes mechanosensitive and activation channels reach steady state quickly, based on timescale separation.
ad hoc to paper Phenomenological expansion of p
Eq. (9) expands polarization as p1 ũ + p3 ũ^3 + p5 ũ^5 without microscopic derivation.
domain assumption Advection-diffusion feedback proportional to migration velocity
Polarity cues are advected by retrograde flow proportional to migration velocity, from prior models (Refs. 5, 32).
domain assumption Anisotropy is small (p << 1)
Assumes the polarization anisotropy is small enough not to affect spreading dynamics.
domain assumption δ1 > 0 and δ2 > 0
Supported by experiments showing volume loss on spreading and RVI on activation.
domain assumption Cl- is passive and fast
Uses steady-state Cl- to relate volume to ψ, neglecting Cl- transport timescale.

invented entities (1)

Polarity cues independent evidence
purpose: Generate spatially inhomogeneous active forces along the contact ring, creating polarization and directional migration.
Generic placeholders for known proteins (IRSp53, arpin, Rac1) cited in the text, so they are not fabricated entities.

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Cite this review

Pith. "Pith review of Theory of cell size regulation during migration in adhered cells." pith.science (2026). https://pith.science/paper/WEUCJQAA

@misc{pith2026250516750,
  author       = {Pith},
  title        = {Pith review of: Theory of cell size regulation during migration in adhered cells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WEUCJQAA}},
  note         = {Machine review of arXiv:2505.16750}
}

read the original abstract

Cell migration is closely linked to cell shape, yet cell size is often assumed to remain constant. This assumption is challenged by recent experiments showing that cells undergo volume loss during spreading and swelling upon activation, with migration velocity correlated to cell size. In this Letter, we present a minimal theoretical framework for cellular size regulation and its influence on migration velocity. We connect cell size to membrane potential and active, actin-driven forces. Spatial inhomogeneities in these forces establish cell polarization and drive migration. Crucially, inhomogeneity is easier to establish over larger sizes, giving rise to a critical contact area, above which migration is possible. Our theory captures the coupled dynamics of cell volume, surface area, and motility and explains recent experiments on neutrophils.

Figures

Figures reproduced from arXiv: 2505.16750 by the authors.

**Figure 2.** FIG. 2: Spreading and swelling dynamics, with and without RVI [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Explicitly, the velocity is given by u =    3 3/2 2 uM Rc R q 1 − Rc R , p3 < 0, R > Rc u ∗ c R ∗ c R s 1 + r Rc Rc−R∗ c 1 − R∗ c R , p3 > 0, R > R∗ c , (10) and vanishes for lower values of the radius. In the case p3 < 0, uM is the maximal velocity, obtained for R = 3Rc/2. For p3 > 0, the velocity suddenly jumps to u = u ∗ c for R = R∗ c . There are two branches: a stable branch that increases with R unt… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: Comparison of theoretical predictions (black dashed curves) with experimental data by Nagy et al. [22] (green curves), for neutrophil [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: Correlation between Volume speed and area (data taken [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 7, 2026 · model on record in the stance chip above.

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