Loops versus lines and the compression stiffening of cells

A. E. Patteson; Anne van Oosten; J. M. Schwarz; Katarzyna Pogoda; M. C. Gandikota; P. A. Janmey; T. A. Engstrom

REVIEW 3 major objections 4 minor 65 references

Loops versus lines and the compression stiffening of cells

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Mouse embryonic fibroblasts compression stiffen under uniaxial compression, and a single semiflexible polymer loop model captures the measured stress-strain curve up to about 35% strain.

desk verdict New AFM compression result and two new model mechanisms, but the stress conversion is unvalidated, so the central claim needs a control experiment. read the letter →

arxiv 1908.03725 v2 pith:L2AHXWOU submitted 2019-08-10 physics.bio-ph cond-mat.softq-bio.CB

M. C. Gandikota , Katarzyna Pogoda , Anne van Oosten , T. A. Engstrom , A. E. Patteson , P. A. Janmey , J. M. Schwarz This is my paper · ORCID

classification physics.bio-phcond-mat.softq-bio.CB

keywords compressionstiffeningsingle-cellmechanicsmouseembryonicfibroblastsatomicforcemicroscopysemiflexiblepolymerloopactomyosincortexarea-conservingloopsfibernetwork

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

Single living cells are generally expected to soften when squeezed, because the semiflexible biopolymer networks that give them structure buckle under compression. This paper reports the opposite for mouse embryonic fibroblasts: atomic force microscopy compression shows their compressive modulus rising with strain, with stiffening beginning around 20% strain. To explain the result, the authors model the cell as a semiflexible polymer loop, the actomyosin cortex, enclosing an incompressible viscous interior, and show that this single-loop model matches the measured stress-strain curve up to roughly 35% compressive strain with one free dimensionless parameter. They also show that fiber networks containing area-conserving loops, representing fluid organelles and vesicles, stiffen by forcing fibers to bend around the loops, and that the stress-strain response is sensitive to the loops' packing fraction and size distribution. The finding matters because it identifies a fast, non-remodeling mechanical response that protects cells under compression and offers a possible building-block mechanism for tissue compression stiffening.

What carries the argument

The central object is the semiflexible polymer loop Hamiltonian $$H_{v+c}=\frac12 K_{cf}\sum_{\langle ij\rangle}(l_{ij}-l_0)^2+\frac12 K_{sf}\sum_{\langle ijk\rangle}(\theta_{ijk}-\theta_0)^2+\$\lambda$(A-A_0),$$ a closed chain of stretchable springs and bending springs whose enclosed area is held fixed by a Lagrange multiplier. The area constraint models an incompressible viscous cytoplasm and supplies an effective outward pressure that prevents the loop from buckling in plane, which is why the loop compression stiffens instead of softening. The dimensionless ratio $\tilde{\kappa}=K_{cf}l_0^2/K_{sf}$ controls whether stretching or bending dominates: as $\tilde{\kappa}\to 0$ the stress grows as $\gamma^3$ with no linear regime, while finite bending produces the linear small-strain response and delayed stiffening seen in the mEF data at $\tilde{\kappa}=0.768$. Transplanting the same loop concept into a two-dimensional fiber network, as area-conserving inclusions and as angular constraints at the mesh scale, generates the bending-driven stiffening mechanisms that apply when a bulk cytoskeletal network or an in vitro fibrin network is present.

What would settle it

Compress mouse embryonic fibroblasts after selectively disrupting the actomyosin cortex, for instance with an actin-depolymerizing drug, leaving the interior intact: the single-loop model predicts that compression stiffening should be abolished or shifted to substantially larger strains, whereas survival of stiffening would show that a bulk fiber network is load-bearing.

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Extended reading notes

Core claim

Mouse embryonic fibroblasts compression stiffen under uniaxial compression imposed by an atomic force microscope, with stiffening setting in at $\gamma_c \approx 20\%$ compressive strain, in contrast to the compression softening expected from semiflexible biopolymer networks. The paper's central quantitative claim is that, up to about 35% compressive strain, the measured stress-strain curve is well described by a single semiflexible polymer loop model of the actomyosin cortex enclosing an incompressible fluid, with only one free parameter, $\tilde{\kappa} = 0.768$, the ratio of stretching to bending stiffness of the loop. In this model, the incompressible interior acts as an outward pressure that suppresses in-plane buckling, so the loop stiffens rather than collapses; with finite bending stiffness the stress-strain curve has a linear small-strain regime followed by nonlinear stiffening, as observed. The paper additionally establishes two bulk-network mechanisms for compression stiffening: area-conserving loops, modeling fluid organelles and vesicles, interspersed in a fiber network force fibers into non-affine bending and convert the network's compression softening into stiffening even at small packing fractions, and angularly constrained crosslinks distort under compression to produce stiffening without organelles. For fibrin networks embedded with adherent beads, the area-conserving-loop model captures the stiffening and stress magnitude, with a later onset of stiffening in the model than in the experiment.

Load-bearing premise

The quantitative fit assumes that a squeezed cell behaves like independent flat slices of an incompressible fluid wrapped in a cortex, with no load-bearing fiber network spanning the cell's interior.

Editorial extensions

If this is right

Single cells can stiffen under compression on short timescales, without waiting for cytoskeletal reorganization, giving them a fast mechanical defense against large homeostatic pressures.
For cortex-dominated cells, uniaxial compression curves collapse onto a one-parameter family; measuring $\tilde{\kappa}$ from a compression test gives a readout of the balance between cortex stretching and bending.
Only a small number of fluid-like organelles or vesicles, with packing fraction around 0.04, can flip a fiber network from compression softening to compression stiffening by forcing fibers to bend.
Because the stress-strain curve depends on the packing fraction and size distribution of area-conserving loops, compression rheology can distinguish cell types or states by their internal organelle geometry.
Cellular compression stiffening is a plausible contributor to tissue compression stiffening, consistent with the observation that decellularized tissue loses much of its stiffening response.

Reading between the lines

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

Editorial inference: if the single-loop description is the operative mechanism, the onset strain $\gamma_c$ and fitted $\tilde{\kappa}$ should shift systematically when cortex composition is altered, so compression tests on cells with more or less actin-crosslinking protein would provide a direct check.
Editorial inference: the mechanical-fingerprint result suggests compression tests could serve as a label-free assay for organelle size and vesicle content; changing organelle size osmotically should move $\gamma_c$ and the stress magnitude in a way the area-conserving-loop model can predict.
Editorial inference: the paper's two-dimensional-to-three-dimensional argument implies a direct three-dimensional loop-shell model should produce quantitatively different stress at large strain; comparing enucleated cells with whole cells would isolate whether the deviation above 35% strain comes from the nucleus or from the percolating bulk network.

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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

The paper reports a new AFM finding—mouse embryonic fibroblasts compression stiffen under spherical-indenter compression—and it offers two new model mechanisms for compression stiffening: area-conserving loops embedded in a fiber network, and angle-constraining crosslinks. That is a real contribution. The modeling is careful, the analytic calculations in the appendices are credible, and the authors are unusually honest about what they cannot rule out.

The thing to know before reading closely is that the experimental stress–strain curve, which anchors the whole paper, is built on a stress conversion that is not validated. The AFM data are converted to stress as σ = F/(2πrh), where r is the sphere radius and h is indentation depth. That is the area of the indenter cap, not the actual contact area for a compliant rounded cell squeezed between a sphere and a dish. For a linear-elastic half-space, Hertz contact gives F ∝ h^{3/2} and therefore σ ∝ h^{1/2}—a concave-down curve with decreasing slope. The paper's concave-up stress–strain curve could be real material stiffening, but it could also come from the area assumption or from finite-thickness/confinement effects. There is no control run with the same AFM sphere and the same analysis on a passive linear-elastic material. The PAA-gel control in Appendix B3 is a rheometer plate compression, not the AFM geometry, so it does not settle this.

The loop-model comparison in Figure 6 is suggestive, not conclusive. It uses one fitted stiffness ratio and normalizes by the experimental pre-stress; that can absorb a fair amount of mismatch. The authors also assume away a bulk cytoskeletal network—Section V says 'let us assume there is not'—and they explicitly say the other two mechanisms cannot be ruled out. Those are limitations, not fatal flaws, but they keep the paper at the level of a plausible mechanism rather than a demonstrated one.

Where the paper earns its keep: the mEF measurement is new, the two network mechanisms are new, and the analytic small-strain calculations in the appendices give the reader a way to check the model behavior independently. The fibrin-bead comparisons are a good attempt to test the organelle-loop idea in a controlled system.

I would send this to a serious referee. The novelty and modeling effort justify the time. But the referee should require either a control experiment that validates the stress conversion or a much more careful treatment of the contact mechanics before the central claim is taken as established. I would not cite it in my own work until that is resolved.

Referee Report

3 major / 4 minor

Summary. The manuscript reports AFM uniaxial compression experiments on rounded mouse embryonic fibroblasts, finding a compressive stress–strain curve that bends upward with stiffening onset near γc ≈ 20%. To explain this, the authors study three mechanical models: (i) a two-dimensional semiflexible polymer loop enclosing an incompressible area, representing the actomyosin cortex with viscous interior; (ii) a diluted triangular fiber network with embedded area-conserving loops, representing organelles and vesicles; and (iii) a fiber network with angle-constraining crosslinks. They report that the single-loop model matches the mEF experiment up to about 35% strain with one dimensionless parameter κ̃, and that the fiber-network-with-loops model captures compression stiffening observed in fibrin networks with adherent beads. Analytical calculations in the appendices support the numerical energy minimizations.

Significance. If the mEF result is a genuine material response, it establishes single-cell compression stiffening on fast timescales, in contrast to expectations from semiflexible polymer networks, and it would be directly relevant to tissue-scale compression stiffening. The study's strengths include explicit numerical simulations with error bars and system-size checks, analytical treatments of the loop model in Appendix A, a separate fibrin-bead experiment that provides a falsifiable test of the area-conserving-loop mechanism, and a PAA-gel control in Appendix B3 showing no stiffening when fiber bending is negligible. However, the load-bearing experimental stress conversion and the number of effective fitting parameters in the central model–experiment comparison must be addressed before the quantitative claims are secure.

major comments (3)

[Section II (Whole cell compression)] The reported stress is computed as σ = F/(2πrh), with r = 12.5 µm and h the indentation depth, while the strain is ε = h/H0. This makes the reported stress proportional to F(ε)/ε. For a linear-elastic half-space indented by a sphere, Hertz contact gives F ∝ ε^{3/2} and hence the reported stress would scale as ε^{1/2}, which has decreasing slope. The observed upward curvature could therefore arise from the assumed contact-area model, finite-thickness or confinement effects, or genuine material stiffening. The PAA-gel control in Appendix B3 uses rheometer plate compression, not the AFM spherical-indenter geometry, so it does not validate the stress conversion used for the cells. A control with the identical AFM sphere protocol and the same σ = F/(2πrh) analysis applied to a passive linear-elastic material of comparable size is needed to support the paper's central experimental claim.
[Section V, Fig. 6] The text states that the loop-model comparison has 'only one free parameter, κ̃,' but the experimental curve is transformed by subtracting the pre-stress and normalizing the stress by that same value. The subtracted offset and the stress scale are determined from the data being explained, and the model stress is separately normalized to the experimental scale. Therefore the comparison effectively uses more than one fitted quantity, and the agreement in Fig. 6 is a weaker test than claimed. The authors should either report the fit with a pre-specified offset and scale, or explicitly count the offset and scale as fitted parameters when assessing the agreement.
[Section V (Comparison with Experiments)] The selection of the single-loop cortex model as the explanation up to 35% strain rests on the stated assumption: 'Since we do not know directly whether or not there is a bulk, rigid cytoskeletal network, let us assume there is not.' This assumption is load-bearing: if a spanning fiber network contributes appreciably to the mEF compression response, the fitted κ̃ and the loop-model agreement could be coincidental. The manuscript does not present any experimental test of this assumption, such as comparing wild-type cells with vimentin-null or actin-disrupted cells under the same compression protocol. Given that the authors themselves identify this as an unknown, the conclusion that the cortex is the dominant contributor up to 35% strain should be either explicitly tested or substantially weakened.

minor comments (4)

[Section IV] In the paragraph following the definition of Ho+fn, the text says 'See Fig. 3b for the stress-strain curves' when discussing the fiber network without loops; the relevant panel appears to be Fig. 4b, not Fig. 3b.
[Section II] The sentence 'The data was obtained from 10 cells and averaged over with the error bar denoting the standard deviation' should be rephrased as 'Data were averaged over 10 cells, with error bars denoting the standard deviation.'
[Section VI] There are two typographical issues: 'mouse embryonic fibroblastas' should be 'mouse embryonic fibroblasts,' and 'experimental experimental techniques' contains a duplicated word.
[Appendix B1 and Fig. 4b] The caption of Fig. 4b states that the analytical fits 'are scaled here to fit with the numerical data,' so the comparison between the numerical network compression and the analytical model is not parameter-free; the scaling factor should be reported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the loop-model stiffening is derived from its own Hamiltonian, and the experimental compression-stiffening claim is independent of the model.

full rationale

The central experimental claim rests on AFM force-distance data converted via sigma = F/(2*pi*r*h) and epsilon = h/H0; whatever the merits of that spherical-cap area normalization, it is a data-analysis convention and not an input to the loop model. The single-loop model's stress-strain curves are obtained by numerical minimization of Eq. (1), with stiffening following from the area-conservation constraint and bending energy (Eqs. A3-A4); no experimental curve is used to define the Hamiltonian. The comparison in Fig. 6 does fit the dimensionless parameter kappa-tilde = 0.768 and normalizes by the experimental pre-stress, so the 'agreement' is a one-parameter fit rather than a parameter-free prediction; but the paper labels kappa-tilde as a free parameter and does not claim the prediction is unconditional. The self-citations ([3] and [20]) supply background or standard percolation values and are not load-bearing. The paper explicitly declines to rule out alternative mechanisms ('we cannot necessarily rule out any of the compression stiffening mechanisms') and admits fluid flow cannot yet be excluded, which further indicates the derivation is not being forced. Concerns about the spherical-cap stress normalization are correctness risks, not circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The models are plausible but rely on several unmeasured assumptions. The 'one free parameter' cortex-loop fit actually uses two adjustments: κ̃ and a pre-stress normalization. The network model requires bending comparable to stretching and area-conserving loops, conditions not directly verified in the mEF experiments. No new physical entities are postulated; area-conserving loops and angle-constraining crosslinks are proxies for known cell components and are treated as modeling constructs.

free parameters (4)

κ̃ (Kcf l0^2 / Ksf) = 0.768
Chosen to match the mEF stress-strain curve in Fig. 6. It controls the onset of nonlinearity in the cortex-loop model and is described as somewhat constrained by earlier experiments, but it is not derived from an independent measurement.
Experimental pre-stress offset and stress scale = Not tabulated; taken from the mEF curve in Fig. 1
In Fig. 6 the authors subtract the pre-stress from the experimental data and use the same value to nondimensionalize stress. This is a second adjustment in the model-experiment comparison beyond κ̃.
Area-spring stiffness ratio KA l0^2 / Kcf = 10^3
Chosen by hand in Section IV to make loops effectively area-conserving in the fiber network. The area-conserving-loop stiffening mechanism depends on this choice.
Rigid-loop stiffness amplification = 100 times Kcf for central-force springs surrounding loops
Chosen in Section V to model rigid dextran beads in the fibrin comparison. It shifts the model's stiffening onset and is not derived from independent measurement.

assumptions (6)

domain assumption A semiflexible loop enclosing an incompressible fluid does not buckle in plane because area conservation pushes the perimeter outward.
Section III: 'We do not consider buckling in our model since semiflexible polymer loops with area conservation acting as a pressure, pushes the perimeter outwards.' If buckling occurred, the loop-model stiffening could be overestimated.
domain assumption The mEF can be represented as independent two-dimensional cross-sections with minimal fluid flow between them during AFM compression.
Section V converts three-dimensional volume conservation to two-dimensional area conservation to justify a quantitative comparison with the 2D loop model.
ad hoc to paper There is no bulk, rigid cytoskeletal network dominating mEF mechanics.
Section V states: 'Since we do not know directly whether or not there is a bulk, rigid cytoskeletal network, let us assume there is not.' The loop-model fit relies on this unverified assumption.
domain assumption Two-dimensional stress can be compared to three-dimensional stress up to a length-factor rescaling.
Section V argues that energy has the same units in any dimension while stress differs by a length factor, and uses this to justify quantitative agreement with the 3D experiment.
domain assumption Fibrin fibers are in the regime where bending energy is comparable to stretching energy.
Section V and the Discussion rely on this to explain why the network-with-loops model applies to fibrin. If the regime is wrong, the proposed bending-driven stiffening mechanism weakens.
domain assumption A randomly diluted triangular lattice with open boundaries and no prestress captures the generic mechanics of the cell cytoskeleton and fibrin networks.
Section IV invokes prior 2D modeling results to justify the lattice choice, but the specific disorder realization and lack of prestress are model inputs not directly measured in the experiments.

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Pith. "Pith review of Loops versus lines and the compression stiffening of cells." pith.science (2026). https://pith.science/paper/L2AHXWOU

@misc{pith2026190803725,
  author       = {Pith},
  title        = {Pith review of: Loops versus lines and the compression stiffening of cells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L2AHXWOU}},
  note         = {Machine review of arXiv:1908.03725}
}

read the original abstract

Both animal and plant tissue exhibit a nonlinear rheological phenomenon known as compression stiffening, or an increase in moduli with increasing uniaxial compressive strain. Does such a phenomenon exist in single cells, which are the building blocks of tissues? One expects an individual cell to compression soften since the semiflexible biopolymer-based cytoskeletal network maintains the mechanical integrity of the cell and in vitro semiflexible biopolymer networks typically compression soften. To the contrary, we find that mouse embryonic fibroblasts (mEFs) compression stiffen under uniaxial compression via atomic force microscopy (AFM) studies. To understand this finding, we uncover several potential mechanisms for compression stiffening. First, we study a single semiflexible polymer loop modeling the actomyosin cortex enclosing a viscous medium modeled as an incompressible fluid. Second, we study a two-dimensional semiflexible polymer/fiber network interspersed with area-conserving loops, which are a proxy for vesicles and fluid-based organelles. Third, we study two-dimensional fiber networks with angular-constraining crosslinks, i.e. semiflexible loops on the mesh scale. In the latter two cases, the loops act as geometric constraints on the fiber network to help stiffen it via increased angular interactions. We find that the single semiflexible polymer loop model agrees well with our AFM experiments until approximately 35% compressive strain. We also find for the fiber network with area-conserving loops model that the stress-strain curves are sensitive to the packing fraction and size distribution of the area-conserving loops, thereby creating a mechanical fingerprint across different cell types. Finally, we make comparisons between this model and experiments on fibrin networks interlaced with beads as well as discuss the tissue-scale implications of cellular compression stiffening.

Figures

Figures reproduced from arXiv: 1908.03725 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** b and Appendix A) that makes an excellent fit to [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. A rhombus 4-gon compressed along the vertical [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10. Collapse of springs induces softening. [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 16.** Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

**Figure 15.** Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]

**Figure 17.** Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

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Compressing a 4-gon We assume the simplest loop symmetric about the x−axis and y−axis, a 4-gon that will be vertically and uniaxially compressed. We choose the arms of the loop to be oblique to the vertical compression rather than to have the arms perfectly parallel to the dir...
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An ellipse in the continuum limit When the number of vertices in the loop is large and ˜κ < 1, then numerical minimization yields elliptical shapes with compressive strain. The continuum limit loop is then assumed to be a circle at zero strain and an ellipse at ﬁnite strain. F...
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(A14) The contour derivative of the unit tangent vector can be expressed in terms of the parameter θ as, dˆt ds = 1 r(θ) dˆt dθ. (A15) Eq. (A11) can now be presented as, Ho+c,sf = κ 2 ∫ 2π 0 dθ r(θ) ⏐⏐⏐⏐ d dθ (−b sin(θ),a cos(θ)) r ⏐⏐⏐⏐ 2 . (A16) Having a,b as functions of ϵ, ...
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Soft area constraint We now study the eﬀect of replacing the Lagrange mul- tiplier term in Eq. 1 with a soft area constraint, i.e. κA(A−A0)2. For small enough KA, the area of the semi- ﬂexible polymer loop can change and so we ask whether or not compression stiﬀening will be o...
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No organelles: Compression softening FIG. 10. Collapse of springs induces softening. We ﬁrst present an approximate calculation for the compression softening mechanism in the absence of or- ganelles (area-conserving loops). The energy of a single central force spring is E = Kc...
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11a) in the vertical direction

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Experiment with polyacrylamide gel To study the eﬀect of bending in the ﬁber network on compression stiﬀening, we study beads embedded in a polyacrylamide (PAA) gel, which is a linear elastic ma- terial. The experimental protocol is the following: 8% acrylamide and 0.3% bis-ac...

2000

Pith tools

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

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obstacle

Compression softening has earlier been observed in a tensegrity model of a cell [37]. What can we expect when we include organelles as area-conserving loops into the ﬁber network? While working with our initial cell as a viscous medium sur- rounded by an actomyosin cortex, we saw that despite a loop of central-force springs being ﬂoppy according to Maxwel...

work page

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When the ratio is small, Hf n+axlinks reduces to Ho+f n (with KA = 0) and compression softening is ob- served as expected

is shown for systems with occupation proba- bilityp = 0.58. When the ratio is small, Hf n+axlinks reduces to Ho+f n (with KA = 0) and compression softening is ob- served as expected. The onset of nonlinearity is not tunable by this ratio. Bottom: With Kxlink/(Kcfl2

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For both ﬁgures, the curves are averaged over 1000 runs on an 8x8 lattice with Kcfl2 0/Ksf = 1

= 0 .1, com- pression stiﬀening for diﬀerent occupation probabilities p is shown. For both ﬁgures, the curves are averaged over 1000 runs on an 8x8 lattice with Kcfl2 0/Ksf = 1. area-conserving loops, this network compression stiﬀ- ens as can be inferred from Fig. 5. At small strain, the angles between ﬁbers change within each triangu- lar loop with both ...

work page

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closer to unity. Note that we can explore a larger range of occu- pation probabilities than with freely-rotating crosslinks because thep above which the network is rigid is the con- nectivity percolation threshold for the triangular lattice, i.e. pc = 2 sin(π

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