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REVIEW 3 major objections 4 minor 57 references

The multiscale mechanics of axon durotaxis

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that a single traction–stiffness curve, derived from molecular clutches and integrated over the growth cone, determines whether an axon is steered by stiffness and that this mechanism produces attractive stiffness lines…

desk verdict A serious three-scale model of axon durotaxis with genuine new content; worth refereeing, but the predictions hang on assumptions that need sensitivity analysis. read the letter →

arxiv 2501.01868 v1 pith:FKOSP4CP submitted 2025-01-03 physics.bio-ph

Christoforos Kassianides , Alain Goriely , Hadrien Oliveri This is my paper · ORCID
classification physics.bio-ph MSC 92C1074L1592C1774K10
keywords axonguidancedurotaxisgrowthconemolecularclutchmorphoelasticrodstiffnessgradientmechanotaxisaxonaloptics
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

During development, growing axons are guided not only by chemicals but by the stiffness of the tissue they grow through—durotaxis. This paper builds a three-scale mechanical theory of axon durotaxis starting from molecular clutches that link actin filaments to the substrate, integrating their forces over the growth cone, and then treating the whole axon as a growing, bendable rod. The central claim is that the sign of the derivative of the traction–stiffness curve $T(\chi)$ decides whether an axon moves up or down a stiffness gradient, and that because this curve can be nonmonotonic, both positive and negative durotaxis can arise from one mechanism. As a consequence, lines of optimal stiffness act as attractors for axon trajectories, and axons hitting a straight stiffness interface behave analogously to light rays—refracting across it or reflecting off it, though without time-reversal symmetry. The model is tested against measurements of the optic tract in Xenopus, where it reproduces the observed deflection of retinal axons and predicts how myosin contractility tunes guidance success.

What carries the argument

The load-bearing object is the traction–stiffness function $T(\chi)$ obtained from a mean-field molecular-clutch model. A rigid actin filament drags at velocity $V$ over a substrate while cross-bridges attach unstretched and detach according to a Bell law $p_{\rm off}\propto e^{f/f_0}$, and all bridges share a single substrate spring of stiffness $\kappa'$; the balance yields $T(\chi)=f_0 K H(u)$ where $\chi=\kappa'/\kappa$ is dimensionless substrate stiffness, $H$ is the special function $H(x)=x e^x\int_0^\infty y\,e^{-x e^y}\,dy$, and $u$ is fixed by a two-equation system. This curve has a global maximum at the optimal stiffness $\chi^*$, expressible in closed form with the incomplete gamma function, and a curvature $T''(\chi^*)$ that sets how sharply axons react. Integration over a circular-sector growth cone gives a mesoscopic force whose leading terms are $2R\rho\sin\varphi_0\,T(\chi)\,\mathbf t + R^2\rho\varphi_0\,T'(\chi)\,\mathbb{P}(\varphi_0)\nabla\chi$, so the sign of $T'$ sets the direction of steering. The axon is then a morphoelastic rod with exponential growth law and a moving clamped tip, reduced to a beam equation for the active distal segment.

What would settle it

Measure the traction–stiffness curve of isolated growth cones on calibrated substrates spanning both sides of the predicted optimal stiffness $\chi^*$ and compare the sign of $T'(\chi)$ with the turning direction in a linear stiffness gradient; if axons on stiffer-than-optimal substrate still turn up-gradient, the central claim fails.

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Extended reading notes

Core claim

The paper's central discovery is that the direction and strength of axonal durotaxis are controlled by the local slope $T'(\chi)$ of the traction generated by a single actin filament as a function of substrate stiffness, and that this function is inherently nonmonotonic under Bell-type adhesion kinetics and a deformable substrate. When the local stiffness is below the optimal stiffness $\chi^*$ where traction is maximal, axons exhibit positive durotaxis; above it, negative durotaxis. Because the traction peaks at $\chi^*$, the stiffness landscape has attractive lines: an axon near the line of optimal stiffness experiences a restoring force that pulls it back toward the line, while sufficiently oblique crossings escape. At a sharp interface between two uniform stiffness regions, the growth cone's finite size generates either refraction or reflection depending on the ratio of tractions and the incidence angle, with no simple Snell's law and no time-reversibility. In a realistic stiffness field from the developing Xenopus brain, the model shows that reflection and refraction at internal interfaces, rather than attraction to optimal-stiffness lines, best account for guiding axons to the optic tectum.

Load-bearing premise

The load-bearing premise is that the growth cone's net steering force is just the sum of independent edge forces from actin filaments that do not interact mechanically and act only at the lamellipodium rim.

Editorial extensions

If this is right

  • On substrates softer than the optimal stiffness, axons can only turn toward stiffer regions; on stiffer substrates, only toward softer regions.
  • If the traction–stiffness curve is monotonic (myosin contraction below a critical value $M_{\rm crit}$), negative durotaxis is impossible; positive durotaxis is the only mode.
  • The line of optimal stiffness in a graded field is a local attractor: nearby axons with small inclination converge to it, whereas sufficiently steep crossings escape.
  • At a straight stiffness interface, axons can refract into the stiffer traction region or reflect back, with reflection angles not equal to incidence angles.
  • In the Xenopus optic-tract scenario, the success rate of reaching the tectum is a nonmonotonic function of myosin contractility $M$, with the best guidance arising from reflection and refraction at internal interfaces rather than tracking optimal-stiffness contours.

Reading between the lines

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

  • Editorial extension: the same nonmonotonic traction–stiffness mechanism could apply to other cell types that use molecular clutches, so durotaxis in fibroblasts or cancer cells may also switch sign around an optimal matrix stiffness.
  • Editorial extension: the optical analogy suggests a design principle—by patterning stiffness landscapes, one could steer growing axons or other cells along prescribed paths, but the absence of time-reversal means such 'axonal lenses' are intrinsically directional.
  • Editorial extension: the prediction of attraction to optimal stiffness lines could be tested directly in vitro by culturing growth cones on stiffness gradients that bracket $\chi^*$ and measuring the distribution of final trajectories; the paper does not report such an experiment.
  • Editorial extension: because the tip-growth approximation assumes a short active zone, the model may underestimate long-range mechanical coupling along the axon; testing with full-axon simulations where the friction lengthscale is not small would reveal whether reflection is robust.
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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 develops a three-scale mechanical model of axonal durotaxis. At the molecular scale, a Bell-type molecular-clutch model of a single actin filament on a deformable substrate yields a traction-stiffness relation T(χ) that can be non-monotonic, with an optimal stiffness χ* if myosin contractility M exceeds a critical value. At the growth-cone scale, the traction is integrated over a circular-sector lamellipodium, and a slow-gradient expansion gives a durotactic force proportional to T′(χ)P(φ0)∇χ, so the sign of durotaxis is set by the sign of T′(χ). At the cellular scale, the axon is modelled as a morphoelastic rod with tip growth, friction, and a growth-cone end load. The model is used to predict reflection and refraction at a straight stiffness interface, attraction to lines of optimal stiffness in a linear gradient, and a qualitative comparison with Xenopus retinal axon guidance. The paper is clearly written and the derivations are mostly explicit, but the headline predictions rest on several strong simplifications, especially the edge-only adhesion assumption in the growth-cone force law.

Significance. If the model's assumptions hold, this is a valuable contribution: it connects molecular clutch mechanics to axon-scale trajectories, gives closed-form expressions for the optimal stiffness and the durotactic sensitivity, and introduces the appealing 'axonal optics' concepts of reflection, refraction, and focalisation. The qualitative reproduction of the Xenopus growth patterns and the non-monotonic dependence of success rate on myosin contractility are suggestive. The paper is honest about its limitations and does not overclaim quantitative biological accuracy. However, the central predictions are not yet backed by robustness tests, and one of the two main physical assumptions (edge-localized, non-interacting adhesion) is load-bearing for the reflection/refraction phenomenology and for the attractive-line result. With additional sensitivity analysis and a finite-width adhesion test, the framework could become a strong foundation for further work.

major comments (3)
  1. [Section 3, Eq. (31)] The growth-cone force is evaluated as an integral of the single-filament traction T(χ) over the lamellipodial edge only: F = ∫ ρ T(χ(r+Reφ)) eφ R dφ. This edge-localized, non-interacting-filament assumption is load-bearing for the interface reflection/refraction diagram in Section 5.1 (Fig. 7) and for the restoring-force expansion in Section 5.2 (Eqs. 56–57). In real growth cones, adhesion and force generation are distributed over the interior of the lamellipodium, so the stiffness field is sampled over an area rather than a one-dimensional rim. Averaging over the cone interior would smooth the tanh interface (Eq. 52) over a length comparable to R and would reduce the transverse force that produces the reflective branch. The manuscript offers no argument, numerical test, or literature estimate showing that Fig. 7(b) survives this averaging. Please provide a distributed-adhesion version of the model, or at least a sensitivity analysis with adhesion patches of finite width, and show how the reflection/refraction phase boundary changes.
  2. [Section 6, Fig. 11] The biological application asserts a non-monotonic success-rate curve and concludes that a combination of reflection and refraction, rather than optimal-stiffness tracking, is the most plausible guidance mechanism. This conclusion rests on a single parameter set (K=0.1, β=f0K, Ec=6000 Pa, R=0.5L, φ0=π/2, βa=0.1) and on a small number of manually chosen M/f0K values (10, 35, 50, >50). No sensitivity analysis over R/L, φ0, βa, Ec, or the initial-condition distribution is reported, and no code or data are provided. Since the success rate includes a catastrophic 0% point (M/f0K=35), it is important to know whether that point and the 89% point are robust to perturbations of these parameters. Please add parameter sweeps and, if possible, release the simulation code.
  3. [Section 5.2, Eqs. (56)–(57)] The claim that the optimal-stiffness line D is 'a locally attracting set' is supported only by the sign of the instantaneous transverse force Fn in Eq. (57) together with the simulations in Fig. 10. However, the axon trajectory is governed by the full free-boundary beam problem (49), where clamp advection, bending resistance, the growth law (51), and the feedback of the force on the shape all enter. A negative Fn for r>0 and θ=0 does not by itself guarantee asymptotic convergence under this dynamics, especially since the same expansion shows Fn depends on θ with a coefficient of the same sign as -θ; the manuscript itself notes escape angles. Either provide a linearized stability analysis of the coupled system (49)–(51) about D, or soften the statement to a local, parameter-dependent tendency.
minor comments (4)
  1. [Fig. 7 caption] The caption is internally inconsistent: panel (a) is described as 'smaller on the right' with α_T=5/4, and panel (b) as 'larger on the right' with α_T=4/5, while the labels in the figure panels show T1>T2. Please correct the caption and ensure that the α_T values and the T1/T2 labels match the actual simulation parameters.
  2. [Eq. (15)] The displayed argument of H appears as qχν(χ+N) or is otherwise ambiguous; from the definition u = q(1+χ^{-1}N)/ν in Eq. (19) and the subsequent equations, T should be f0K H(q(χ+N)/(χν)). Please verify the typesetting and correct the expression.
  3. [Eqs. (19), (46), (54)] The symbol β is used for the dimensionless clutch parameter in Eq. (19), for the dimensionless adhesion parameter in Eq. (46), and then β_a appears in Eq. (54). This overloading is confusing; please rename at least one of these parameters.
  4. [General] No statement of code or data availability is provided. The numerical results in Figs. 7–11 would be considerably easier to assess if the simulation code or a reproducibility description were included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the durotaxis, attraction-to-optimal-stiffness, and reflection/refraction results are derived consequences of the model equations, not restatements of the model inputs.

full rationale

After tracing the derivation chain from the molecular clutch model (Section 2) through the mesoscopic force integration (Eq. 31 and the expansion in Eq. 32) to the macroscopic rod and beam equations, the qualitative predictions are consequences of the model's own mathematics, not circular restatements of its inputs. The traction–stiffness function T(χ) is derived from the kinetic equations (13)–(21), and the existence of a maximum at χ* is itself a derived condition (inequality 24), not an imposed ansatz. The positive/negative durotaxis result follows from the sign of T′(χ) in the expansion (32), but that expansion is derived, and the sign of T′ on either side of χ* is a property of the solved traction curve. The attractive character of the optimal-stiffness line D is obtained from the second-order expansion (56)–(57) using T″(χ*)<0, again a derived property. The reflection/refraction behavior in Section 5.1 is obtained by numerically integrating the same force law across a tanh-smoothed interface, not by assuming the outcome. The biological application uses an independently measured AFM stiffness field and scans myosin contractility M; it does not fit parameters to the target endpoints, and the paper explicitly acknowledges that definitive quantitative conclusions require better parameter estimates. The only self-citations (Oliveri et al. 2021; Oliveri and Goriely 2022) supply standard ingredients such as the tip-growth velocity law and the optics analogy; the current derivations do not reduce to those citations, and independent support exists in O’Toole et al. (2008) and in the external experimental patterns used only for qualitative comparison. No step was found where an output is definitionally equal to an input or where a fitted parameter is renamed as a prediction.

Assumptions & free parameters 10 free parameters · 11 assumptions · 0 invented entities

The central claims rest on a chain of modeling assumptions, most importantly the soft-substrate single-spring molecular clutch picture, the edge-localized growth cone force, and the choice of myosin contractility above Mcrit to create a nonmonotonic traction curve. Several dimensionless parameters (K, beta, q, R/L, phi0, lambda, m_chi) are chosen for simulations rather than measured. No new physical entities are proposed.

free parameters (10)
  • M (myosin contractile force) = M/f0K = 10, 35, 50, 70 in Fig. 11; M=7.5 f0K in Fig. 9
    Controls whether the traction-stiffness curve becomes nonmonotonic (M > Mcrit), which is required for negative durotaxis and for attractive optimal-stiffness lines.
  • K = S0/koff (binding constant) = 0.1
    Dimensionless binding constant in the molecular clutch model; value is chosen for simulations, not measured.
  • beta = koff Xi f0 / kappa (dimensionless friction) = beta = f0K
    Couples myosin, clutch, and cytoplasmic drag; chosen for simulations, not measured.
  • q = f0/(kappa ell) (bond strength) = unspecified
    Dimensionless bond strength in the detachment law; appears in the traction equations and is not estimated.
  • Ec (characteristic pressure) = 6000 Pa
    Used to convert AFM stiffness data to dimensionless chi in the Xenopus application; chosen to match the experimental stiffness range.
  • R/L (growth cone radius relative to bending length) = 0.5
    Sets the magnitude of the durotactic steering force in the mesoscopic expansion; used in all simulations.
  • phi0 (growth cone half-angle) = pi/2
    Shape parameter; with phi0=pi/2 the durotactic force aligns with the stiffness gradient, simplifying the analysis.
  • lambda (interface width) = R/4
    Smoothing length for the sharp interface in the reflection/refraction simulations; chosen small enough that results are insensitive.
  • m_chi (stiffness gradient) = -0.1
    Used in the disk illustration (Fig. 9) to mimic the Xenopus stiffness gradient; chosen for qualitative comparison.
  • n0 (critical tension for growth) = 0
    Neglects the growth threshold in the exponential growth law, simplifying the rod model.
assumptions (11)
  • domain assumption Bell-type detachment law with tension-dependent off-rate (Eq. 2)
    Standard biophysical model for adhesion bond breakage under force; inherited from Bell (1978) and Sens (2013).
  • standard math Lacker-Peskin transport equation for bound cross-bridge extension distribution (Eq. 4)
    Governing equation for the mean-field distribution of cross-bridge extensions; a standard result in adhesion dynamics.
  • domain assumption Soft substrate modeled as a single spring kappa' shared by all cross-bridges (Eq. 8)
    Assumes uniform substrate deformation; neglects nonlocal elastic interactions between adhesion sites which may be important in stiffer substrates.
  • domain assumption Constant pool of free cross-bridges and attachment at zero extension (Eq. 5)
    Used in the mean-field kinetic model; assumes binding is not limited by depletion and newly formed bonds are unstretched.
  • domain assumption Growth cone modeled as a circular sector with edge-localized adhesion only (Eq. 31)
    Assumes all traction arises from the lamellipodium edge and that internal forces and filament interactions are negligible.
  • domain assumption Slow stiffness variation permits a Taylor expansion of the growth cone force to O(R^2) (Eq. 32)
    Assumes the growth cone is small compared to the stiffness gradient length scale; used for the durotactic force expansion.
  • standard math Morphoelastic rod with multiplicative decomposition of stretch (Eq. 39)
    Standard morphoelasticity framework from Goriely (2017) for growing elastic rods.
  • ad hoc to paper Exponential growth law with zero threshold (Eq. 42, n0=0)
    A specific phenomenological growth law; setting n0=0 removes the critical tension threshold, simplifying the model.
  • ad hoc to paper Curvature remodeling law with single timescale (Eq. 43)
    Assumes internal remodeling relaxes bending moment exponentially; timescale tau_gamma is not specified.
  • domain assumption Tip growth approximation with small deflections and a clamped distal segment (Eq. 48-49)
    Assumes the active zone is short and the deflection is small, reducing the rod equations to linear beam equations.
  • ad hoc to paper No mechanical coupling between axons in the biological application
    Assumes individual axons act independently in the optic tract, which the paper presents as a complementary scenario to bundle models.

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Cite this review

Pith. "Pith review of The multiscale mechanics of axon durotaxis." pith.science (2026). https://pith.science/paper/FKOSP4CP

@misc{pith2026250101868,
  author       = {Pith},
  title        = {Pith review of: The multiscale mechanics of axon durotaxis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKOSP4CP}},
  note         = {Machine review of arXiv:2501.01868}
}
read the original abstract

During neurodevelopment, neuronal axons navigate through the extracellular environment, guided by various cues to establish connections with distant target cells. Among other factors, axon trajectories are influenced by heterogeneities in environmental stiffness, a process known as durotaxis, the guidance by substrate stiffness gradients. Here, we develop a three-scale model for axonal durotaxis. At the molecular scale, we characterise the mechanical interaction between the axonal growth cone cytoskeleton, based on molecular-clutch-type interactions dependent on substrate stiffness. At the growth cone scale, we spatially integrate this relationship to obtain a model for the traction generated by the entire growth cone. Finally, at the cell scale, we model the axon as a morphoelastic filament growing on an adhesive substrate, and subject to durotactic growth cone traction. Firstly, the model predicts that, depending on the local substrate stiffness, axons may exhibit positive or negative durotaxis, and we show that this key property entails the existence of attractive zones of preferential stiffness in the substrate domain. Second, we show that axons will exhibit reflective and refractive behaviour across interface between regions of different stiffness, a basic process which may serve in the deflection of axons. Lastly, we test our model in a biological scenario wherein durotaxis was previously identified as a possible guidance mechanism in vivo. Overall, this work provides a general mechanistic theory for exploring complex effects in axonal mechanotaxis and guidance in general.

Figures

Figures reproduced from arXiv: 2501.01868 by the authors.

Figure 1
Figure 1. The multiscale structure of the axon. (a) Molecular scale. An actin filament is pulled [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. General structure of the model. From left to right: microscopic scale–an actin filament [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Structure of the microscopic model. We assume that all cross-bridges (brown springs [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Profile of the traction-stiffness curve T (χ). For small myosin concentration, M < Mcrit, the traction increases monotonically as a function of the stiffness. For large myosin concentration, M > Mcrit, traction has a maximum value at the optimal stiffness χ ∗ . to t. W…
Figure 5
Figure 5. Figure 5: A morphoelastic rod model. Left: At a point [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Schematic of the incremental axonal growth process. We assume that only a segment of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Simulations of axons refracting across and reflecting against an interface with incidence [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Demonstration of the absence of time-reversibility in the system. Approaching the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: (Left) Axons growing from a Xenopus retina, cultured on a substrate with a linear stiffness [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: (a) Axons trajectories converge to the line of optimal stiffness, [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: (a) Success rate as a function of M. (b) Traction as a function of E for the values of M indicated in (a). (c) Density map of traction for the values of M indicated in (a) overlaid with representative trajectories of axons for the given parameters. K = 0.1, β = f0K, T…

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Works this paper leans on

57 extracted references · 47 canonical work pages

  1. [1]

    , year 2000

    author Aeschlimann, M. , year 2000 . title Biophysical model of axonal pathfinding . Ph.D. thesis. University of Lausanne, Department of Theoretical Physics

    work page 2000

  2. [2]

    , author Odde, D.J

    author Bangasser, B.L. , author Odde, D.J. , year 2013 . title Master equation-based analysis of a motor-clutch model for cell traction force . journal Cellular and molecular bioengineering volume 6 , pages 449--459 . https://doi.org/10.1007/s12195-013-0296-5, :10.1007/s12195-013-0296-5

    work page doi:10.1007/s12195-013-0296-5 2013

  3. [3]

    , author Rosenfeld, S.S

    author Bangasser, B.L. , author Rosenfeld, S.S. , author Odde, D.J. , year 2013 . title Determinants of maximal force transmission in a motor-clutch model of cell traction in a compliant microenvironment . journal Biophysical journal volume 105 , pages 581--592 . https://www.cell.com/biophysj/fulltext/S0006-3495(11)00376-6, :10.1016/j.bpj.2013.06.027

    work page doi:10.1016/j.bpj.2013.06.027 2013

  4. [4]

    , author Shamsan, G.A

    author Bangasser, B.L. , author Shamsan, G.A. , author Chan, C.E. , author Opoku, K.N. , author T \"u zel, E. , author Schlichtmann, B.W. , author Kasim, J.A. , author Fuller, B.J. , author McCullough, B.R. , author Rosenfeld, S.S. , et al., year 2017 . title Shifting the optimal stiffness for cell migration . journal Nature Communications volume 8 , page...

    work page doi:10.1038/ncomms15313 2017

  5. [5]

    , year 1978

    author Bell, G.I. , year 1978 . title Models for the specific adhesion of cells to cells . journal Science volume 200 , pages 618--627 . https://www.science.org/doi/10.1126/science.347575, :10.1126/SCIENCE.347575

    work page doi:10.1126/science.347575 1978

  6. [6]

    , author Dembo, M

    author Bell, G.I. , author Dembo, M. , author Bongrand, P. , year 1984 . title Cell adhesion. competition between nonspecific repulsion and specific bonding . journal Biophysical journal volume 45 , pages 1051--1064 . https://www.cell.com/biophysj/pdf/S0006-3495(84)84252-6.pdf, :10.1016/S0006-3495(84)84252-6

    work page doi:10.1016/s0006-3495(84)84252-6 1984

  7. [7]

    , year 2020

    author Bressloff, P.C. , year 2020 . title Stochastic resetting and the mean-field dynamics of focal adhesions . journal Physical Review E volume 102 , pages 022134 . https://journals.aps.org/pre/abstract/10.1103/PhysRevE.102.022134, :10.1103/PhysRevE.102.022134

    work page doi:10.1103/physreve.102.022134 2020

  8. [8]

    , author Odde, D.J

    author Chan, C.E. , author Odde, D.J. , year 2008 . title Traction dynamics of filopodia on compliant substrates . journal Science volume 322 , pages 1687--1691 . https://www.science.org/doi/10.1126/science.1163595, :10.1126/science.1163595

    doi:10.1126/science.1163595 2008

Show all 57 references
  1. [9]

    , author Borowski, P

    author Chaudhuri, D. , author Borowski, P. , author Zapotocky, M. , year 2011 . title Model of fasciculation and sorting in mixed populations of axons . journal Physical Review E—Statistical, Nonlinear, and Soft Matter Physics volume 84 , pages 021908 . https://journals.aps.or...

    2011 doi

  2. [10]

    , year 2007

    author De Gennes, P.G. , year 2007 . title Collective neuronal growth and self organization of axons . journal Proceedings of the National Academy of Sciences volume 104 , pages 4904--4906 . https://www.pnas.org/doi/full/10.1073/pnas.0609871104, :10.1073/pnas.0609871104

    2007 doi

  3. [11]

    , author Lamoureux, P

    author Dennerll, T.J. , author Lamoureux, P. , author Buxbaum, R.E. , author Heidemann, S.R. , year 1989 . title The cytomechanics of axonal elongation and retraction . journal Journal of Cell Biology volume 109 , pages 3073--3083 . https://rupress.org/jcb/article-abstract/109...

    1989 doi

  4. [12]

    , author Trepat, X

    author Elosegui-Artola, A. , author Trepat, X. , author Roca-Cusachs, P. , year 2018 . title Control of mechanotransduction by molecular clutch dynamics . journal Trends in Cell Biology volume 28 , pages 356--367 . https://www.cell.com/trends/cell-biology/fulltext/S0962-8924(1...

    2018 doi

  5. [13]

    , author Marchant, C.L

    author Espina, J.A. , author Marchant, C.L. , author Barriga, E.H. , year 2022 . title Durotaxis: the mechanical control of directed cell migration . journal The FEBS journal volume 289 , pages 2736--2754 . https://febs.onlinelibrary.wiley.com/doi/10.1111/febs.15862, :10.1111/...

    2022 doi

  6. [14]

    , year 2020

    author Franze, K. , year 2020 . title Integrating Chemistry and Mechanics: The Forces Driving Axon Growth . journal Annual Review of Cell and Developmental Biology volume 36 , pages 61--83 . https://www.annualreviews.org/content/journals/10.1146/annurev-cellbio-100818-125157, ...

    2020 doi

  7. [15]

    , author Guck, J

    author Franze, K. , author Guck, J. , year 2010 . title The biophysics of neuronal growth . journal Reports on Progress in Physics volume 73 . https://doi.org/10.1088/0034-4885/73/9/094601, :10.1088/0034-4885/73/9/094601

    2010 doi

  8. [16]

    , year 2017

    author Goriely, A. , year 2017 . title The mathematics and mechanics of biological growth . volume volume 45 of series Interdisciplinary applied mathematics . edition 1st ed., publisher Springer-Verlag, New York . https://link.springer.com/book/10.1007/978-0-387-87710-5, :10.1...

    2017 doi

  9. [17]

    , author Budday, S

    author Goriely, A. , author Budday, S. , author Kuhl, E. , year 2015 . title Neuromechanics: From neurons to brain , in: booktitle Advances in Applied Mechanics . publisher Academic Press Inc. . volume volume 48 , pp. pages 79--139 . https://doi.org/10.1016/bs.aams.2015.10.002...

    2015 doi

  10. [18]

    , author Moulton, D.E

    author Goriely, A. , author Moulton, D.E. , author Mihai, L.A. , year 2023 . title A rod theory for liquid crystalline elastomers . journal Journal of Elasticity volume 153 , pages 509--532 . https://link.springer.com/article/10.1007/s10659-021-09875-z, :10.1007/s10659-021-09875

    2023 doi

  11. [19]

    , author Van Ooyen , A

    author Hentschel, H.G. , author Van Ooyen , A. , year 1999 . title Models of axon guidance and bundling during development . journal Proceedings of the Royal Society B: Biological Sciences volume 266 , pages 2231--2238 . https://doi.org/10.1098/rspb.1999.0913, :10.1098/rspb.1999.0913

    1999

  12. [20]

    , author Miller, K.E

    author Holland, M.A. , author Miller, K.E. , author Kuhl, E. , year 2015 . title Emerging Brain Morphologies from Axonal Elongation . journal Annals of Biomedical Engineering volume 43 , pages 1640--1653 . https://doi.org/10.1007/s10439-015-1312-9, :10.1007/s10439-015-1312-9

    2015 doi

  13. [21]

    , year 2010

    author Hwu, C. , year 2010 . title Infinite Space, Half-Space, and Bimaterials . publisher Springer US , address Boston, MA . chapter chapter 4 . pp. pages 87--113 . https://doi.org/10.1007/978-1-4419-5915-7_4, :10.1007/978-1-4419-5915-7_4

    2010 doi

  14. [22]

    , author Park, K.Y

    author Isomursu, A. , author Park, K.Y. , author Hou, J. , author Cheng, B. , author Mathieu, M. , author Shamsan, G.A. , author Fuller, B. , author Kasim, J. , author Mahmoodi, M.M. , author Lu, T.J. , author Genin, G.M. , author Xu, F. , author Lin, M. , author Distefano, M....

    2022

  15. [23]

    , author Thompson, A.J

    author Koser, D.E. , author Thompson, A.J. , author Foster, S.K. , author Dwivedy, A. , author Pillai, E.K. , author Sheridan, G.K. , author Svoboda, H. , author Viana, M. , author da Fontoura Costa, L. , author Guck, J. , author Holt, C.E. , author Franze, K. , year 2016 . ti...

    2016 doi

  16. [24]

    , author T \'o th, J

    author Kov \'a cs, M. , author T \'o th, J. , author Het \'e nyi, C. , author M \'a ln \'a si-Csizmadia, A. , author Sellers, J.R. , year 2004 . title Mechanism of blebbistatin inhibition of myosin II . journal Journal of Biological Chemistry volume 279 , pages 35557--35563 . ...

    2004 doi

  17. [25]

    , author Buxbaum, R.E

    author Lamoureux, P. , author Buxbaum, R.E. , author Heidemann, S.R. , year 1989 . title Direct evidence that growth cones pull . journal Nature volume 340 , pages 159--162 . http://www.nature.com/articles/340159a0, :10.1038/340159a0

    1989 doi

  18. [26]

    , author Espreafico, E.M

    author Lin, C.H. , author Espreafico, E.M. , author Mooseker, M.S. , author Forscher, P. , year 1996 . title Myosin drives retrograde f-actin flow in neuronal growth cones . journal Neuron volume 16 , pages 769--782 . https://doi.org/10.1016/s0896-6273(00)80097-5, :10.1016/s08...

    1996 doi

  19. [27]

    , author Shinbrot, T

    author Maskery, S. , author Shinbrot, T. , year 2005 . title Deterministic and stochastic elements of axonal guidance . journal Annual Review of Biomedical Engineering volume 7 , pages 187--221 . https://www.annualreviews.org/content/journals/10.1146/annurev.bioeng.7.060804.10...

    2005

  20. [28]

    , author Gupton, S.L

    author McCormick, L.E. , author Gupton, S.L. , year 2020 . title Mechanistic advances in axon pathfinding . journal Current Opinion in Cell Biology volume 63 , pages 11--19 . https://www.sciencedirect.com/science/article/pii/S0955067419301139, :10.1016/j.ceb.2019.12.003

    2020 doi

  21. [29]

    , author Suter, D.M

    author Miller, K.E. , author Suter, D.M. , year 2018 . title An integrated cytoskeletal model of neurite outgrowth . journal Frontiers in cellular neuroscience volume 12 , pages 447 . https://doi.org/10.3389/fncel.2018.00447, :10.3389/fncel.2018.00447

    2018

  22. [30]

    , author Fothergill, T

    author Mortimer, D. , author Fothergill, T. , author Pujic, Z. , author Richards, L.J. , author Goodhill, G.J. , year 2008 . title Growth cone chemotaxis . journal Trends in Neurosciences volume 31 , pages 90--98 . https://doi.org/10.1016/j.tins.2007.11.008, :10.1016/j.tins.20...

    2008 doi

  23. [31]

    , author Oliveri, H

    author Moulton, D.E. , author Oliveri, H. , author Goriely, A. , year 2020 . title Multiscale integration of environmental stimuli in plant tropism produces complex behaviors . journal Proceedings of the National Academy of Sciences volume 117 , pages 32226--32237 . https://do...

    2020 doi

  24. [32]

    , author Hogue, I.B

    author Nguyen, T.D. , author Hogue, I.B. , author Cung, K. , author Purohit, P.K. , author McAlpine, M.C. , year 2013 . title Tension-induced neurite growth in microfluidic channels . journal Lab on a Chip volume 13 , pages 3735--3740 . https://pubs.rsc.org/en/content/articlel...

    2013 doi

  25. [33]

    , author Besser, A

    author Nicolas, A. , author Besser, A. , author Safran, S.A. , year 2008 . title Dynamics of cellular focal adhesions on deformable substrates: consequences for cell force microscopy . journal Biophysical journal volume 95 , pages 527--539 . https://doi.org/10.1529/biophysj.10...

    2008 doi

  26. [34]

    , author De Rooij, R

    author Oliveri, H. , author De Rooij, R. , author Kuhl, E. , author Goriely, A. , year 2022 . title Rheology of growing axons . journal Physical Review Research volume 4 , pages 033125 . https://link.aps.org/doi/10.1103/PhysRevResearch.4.033125, :10.1103/PhysRevResearch.4.033125

    2022 doi

  27. [35]

    , author Franze, K

    author Oliveri, H. , author Franze, K. , author Goriely, A. , year 2021 . title Theory for Durotactic Axon Guidance . journal Physical Review Letters volume 126 , pages 118101 . https://doi.org/10.1103/PhysRevLett.126.118101, :10.1103/PhysRevLett.126.118101

    2021 doi

  28. [36]

    , author Goriely, A

    author Oliveri, H. , author Goriely, A. , year 2022 . title Mathematical models of neuronal growth . journal Biomechanics and Modeling in Mechanobiology volume 21 , pages 89--118 . https://link.springer.com/article/10.1007/s10237-021-01539-0, :10.1007/S10237-021-01539-0

    2022 doi

  29. [37]

    , author Lamoureux, P

    author O'Toole, M. , author Lamoureux, P. , author Miller, K.E. , year 2008 . title A physical model of axonal elongation: Force, viscosity, and adhesions govern the mode of outgrowth . journal Biophysical Journal volume 94 , pages 2610--2620 . https://www.cell.com/biophysj/fu...

    2008 doi

  30. [38]

    , author Miller, K.E

    author O'Toole, M. , author Miller, K.E. , year 2011 . title The role of stretching in slow axonal transport . journal Biophysical Journal volume 100 , pages 351--360 . https://www.cell.com/biophysj/fulltext/S0006-3495(10)05210-0, :10.1016/j.bpj.2010.12.3695

    2011 doi

  31. [39]

    , author Teukolsky, S.A

    author Press, W.H. , author Teukolsky, S.A. , author Vetterling, W.T. , author Flannery, B.P. , year 2007 . title Numerical recipes: The art of scientific computing . edition 3rd ed., publisher Cambridge University Press , address New York . https://www.cambridge.org/de/univer...

    2007

  32. [40]

    , author Smith, D.H

    author Purohit, P.K. , author Smith, D.H. , year 2016 . title A model for stretch growth of neurons . journal Journal of biomechanics volume 49 , pages 3934--3942 . https://www.sciencedirect.com/science/article/pii/S0021929016312246, :10.1016/j.jbiomech.2016.11.045

    2016 doi

  33. [41]

    , year 2023

    author Raffa, V. , year 2023 . title Force: A messenger of axon outgrowth . journal Seminars in Cell & Developmental Biology volume 140 , pages 3--12 . https://doi.org/10.1016/j.semcdb.2022.07.004, :10.1016/j.semcdb.2022.07.004. note special issue: Driving forces behind the wi...

    2023 doi

  34. [42]

    , author J\'erusalem, A

    author Recho, P. , author J\'erusalem, A. , author Goriely, A. , year 2016 . title Growth, collapse, and stalling in a mechanical model for neurite motility . journal Physical Review E volume 93 . https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.032410, :10.1103/PhysR...

    2016 doi

  35. [43]

    , author Schwarz, U.S

    author Sabass, B. , author Schwarz, U.S. , year 2010 . title Modeling cytoskeletal flow over adhesion sites: competition between stochastic bond dynamics and intracellular relaxation . journal Journal of Physics: Condensed Matter volume 22 , pages 194112 . https://doi.org/10.1...

    2010 doi

  36. [44]

    , author Venturini, C

    author S \'a ez, P. , author Venturini, C. , year 2023 . title Positive , negative and controlled durotaxis . journal Soft Matter volume 19 , pages 2993--3001 . https://doi.org/10.1039/D2SM01326F, :10.1039/D2SM01326F

    2023 doi

  37. [45]

    , author Schaffran, B

    author Santos, T.E. , author Schaffran, B. , author Broguière, N. , author Meyn, L. , author Zenobi-Wong, M. , author Bradke, F. , year 2020 . title Axon growth of cns neurons in three dimensions is amoeboid and independent of adhesions . journal Cell Reports volume 32 , pages...

    2020

  38. [46]

    , author Weber, I.P

    author Schaeffer, J. , author Weber, I.P. , author Thompson, A.J. , author Keynes, R.J. , author Franze, K. , year 2022 . title Axons in the chick embryo follow soft pathways through developing somite segments . journal Frontiers in Cell and Developmental Biology volume 10 . h...

    2022

  39. [47]

    , year 2013

    author Sens, P. , year 2013 . title Rigidity sensing by stochastic sliding friction . journal Europhysics Letters volume 104 , pages 38003 . https://iopscience.iop.org/article/10.1209/0295-5075/104/38003, :10.1209/0295-5075/104/38003

    2013 doi

  40. [48]

    , author Mayor, R

    author Shellard, A. , author Mayor, R. , year 2021 . title Durotaxis: The hard path from in vitro to in vivo . journal Developmental Cell volume 56 , pages 227--239 . https://www.sciencedirect.com/science/article/pii/S153458072030928X, :10.1016/j.devcel.2020.11.019

    2021 doi

  41. [49]

    , author Erickson, R.O

    author Silk, W.K. , author Erickson, R.O. , year 1979 . title Kinematics of plant growth . journal Journal of Theoretical Biology volume 76 , pages 481--501 . https://www.sciencedirect.com/science/article/pii/0022519379900146, :https://doi.org/10.1016/0022-5193(79)90014-6

    1979

  42. [50]

    , author Rabbitt, R

    author Smeal, R.M. , author Rabbitt, R. , author Biran, R. , author Tresco, P.A. , year 2005 . title Substrate curvature influences the direction of nerve outgrowth . journal Annals of biomedical engineering volume 33 , pages 376--382 . https://link.springer.com/article/10.100...

    2005 doi

  43. [51]

    , author Fouquet, C

    author S m \' i t, D. , author Fouquet, C. , author Pincet, F. , author Zapotocky, M. , author Trembleau, A. , year 2017 . title Axon tension regulates fasciculation/defasciculation through the control of axon shaft zippering . journal eLife volume 6 . https://elifesciences.or...

    2017 doi

  44. [52]

    , author Walcott, S

    author Srinivasan, M. , author Walcott, S. , year 2009 . title Binding site models of friction due to the formation and rupture of bonds: state-function formalism, force-velocity relations, response to slip velocity transients, and slip stability . journal Physical Review E vo...

    2009 doi

  45. [53]

    , author Miller, K.E

    author Suter, D.M. , author Miller, K.E. , year 2011 . title The emerging role of forces in axonal elongation . journal Progress in Neurobiology volume 94 , pages 91--101 . https://www.sciencedirect.com/science/article/pii/S0301008211000530, :10.1016/j.pneurobio.2011.04.002

    2011 doi

  46. [54]

    , author Pillai, E.K

    author Thompson, A.J. , author Pillai, E.K. , author Dimov, I.B. , author Foster, S.K. , author Holt, C.E. , author Franze, K. , year 2019 . title Rapid changes in tissue mechanics regulate cell behaviour in the developing embryonic brain . journal eLife volume 8 , pages e3935...

    2019 doi

  47. [55]

    , year 2000

    author Trefethen, L.N. , year 2000 . title Spectral methods in MATLAB . publisher SIAM . https://epubs.siam.org/doi/book/10.1137/1.9780898719598, :10.1137/1.9780898719598

    2000 doi

  48. [56]

    , author Bridgman, P.C

    author Turney, S.G. , author Bridgman, P.C. , year 2005 . title Laminin stimulates and guides axonal outgrowth via growth cone myosin II activity . journal Nature neuroscience volume 8 , pages 717--719 . https://www.nature.com/articles/nn1466, :10.1038/nn1466

    2005 doi

  49. [57]

    , year 2008

    author Wolpert, L. , year 2008 . title The French Flag Problems A Contribution to the Discussion on Pattern Development and Regulation , in: editor Waddington, C.H. (Ed.), booktitle The Origin of Life . publisher Routledge . volume volume 1 , pp. pages 125--133 . https://www.t...

    2008 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.