Symmetry-protected phases in a 1D active solid with mechanochemical feedback

Lakshman Santhosh Kumar; Phanindra Dewan; Soumyadeep Mondal; Sumantra Sarkar

arxiv: 2504.19655 · v2 · submitted 2025-04-28 · ❄️ cond-mat.soft · cond-mat.stat-mech· nlin.PS· physics.bio-ph

Symmetry-protected phases in a 1D active solid with mechanochemical feedback

Soumyadeep Mondal , Phanindra Dewan , Lakshman Santhosh Kumar , Sumantra Sarkar This is my paper

Pith reviewed 2026-05-22 19:06 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mechnlin.PSphysics.bio-ph

keywords active solidsmechanochemical feedbacksymmetry-protected phasesoscillation deathHopf oscillatorsself-organizationbiological tissues

0 comments

The pith

Reciprocal coupling of elasticity and Hopf oscillators in a 1D active solid produces symmetry-protected phases and a universal transition to compression-driven oscillation death.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a model of active solids in which mechanical elasticity and chemical oscillators exchange feedback in both directions. Amplitude equations derived from this coupling, combined with group theory, identify a set of phases whose stability is enforced by symmetry. One universal feature is a transition in which applied compression stops the oscillations, offering a mechanism for localized signal dampening in tissues. The classification holds without needing detailed parameter values once symmetry is taken into account.

Core claim

Complex self-organization in active solids with mechanochemical feedback can be classified purely through symmetry arguments, yielding a rich landscape of symmetry-protected phases together with a universal transition to compression-driven oscillation death that supplies a physical account of localized signaling dampening in biological tissues.

What carries the argument

Reciprocal coupling between elasticity and Hopf oscillators, analyzed through amplitude equations and group-theoretic symmetry classification.

If this is right

Symmetry arguments alone suffice to predict the full set of stable phases in such active solids.
The compression-driven oscillation death transition supplies a mechanism for localized dampening of chemical signals inside tissues.
Inconsistencies in earlier models of signaling dampening are resolved by the existence of this universal transition.
Self-organized patterns in active solids follow universal symmetry rules independent of most microscopic details.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same symmetry classification could be extended to two- or three-dimensional active solids to predict additional protected states.
Synthetic active materials could be engineered to exhibit controllable oscillation death for precise spatial signaling.
Experiments that vary compression while tracking oscillation amplitude in real cell chains would directly test the predicted transition.

Load-bearing premise

The reciprocal coupling between elasticity and Hopf oscillators in the one-dimensional model captures the essential mechanochemical feedback present in real biological tissues.

What would settle it

Direct observation that compression applied to a one-dimensional chain of coupled oscillators fails to produce uniform oscillation death, or that the observed phases violate the symmetry-protected pattern predicted by the amplitude equations, would refute the central claim.

Figures

Figures reproduced from arXiv: 2504.19655 by Lakshman Santhosh Kumar, Phanindra Dewan, Soumyadeep Mondal, Sumantra Sarkar.

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

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

We present a framework for mechanochemical self-organization in active solids where elasticity is reciprocally coupled to Hopf oscillators. Our model reveals a rich landscape of symmetry-protected phases, identified through amplitude equations and group-theoretic analysis. We uncover a universal transition to compression-driven oscillation death (COD), providing a physical basis for localized signaling dampening in biological tissues that resolves inconsistencies in previous models. Our work demonstrates that complex self-organization in active solids can be classified purely through symmetry arguments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper couples elasticity reciprocally to Hopf oscillators in a 1D active solid, then uses amplitude equations plus group theory to classify phases and locate a compression-driven oscillation death transition.

read the letter

The main thing to know is that the authors build a 1D lattice model where local Hopf oscillators feed back into the elastic strain and vice versa, reduce the system to amplitude equations, and then apply symmetry arguments to label the steady states. They report a transition to oscillation death under increasing compression that they describe as universal and potentially relevant to localized signaling in tissues. That combination of reciprocal mechanochemical coupling with an explicit symmetry classification is the concrete addition over earlier active-solid or oscillator models. The symmetry route is a clean way to organize the phases once the reduction is in hand, and the link to biological dampening gives the work a clear target audience. On the soft spots, the stress-test concern is worth taking seriously. Amplitude equations assume the oscillators stay near a limit cycle with slowly varying amplitudes, yet the COD point is defined by the amplitude dropping to zero. If the local dynamics simply cross a standard Hopf threshold rather than a symmetry-protected state, the universality claim rests on an unverified approximation. The paper would need to show that the multiple-scale or center-manifold conditions remain satisfied right up to the transition; without that check the COD looks more like an ordinary bifurcation. The reciprocal coupling itself is an assumption that may or may not match real tissue mechanics, though the model is at least internally consistent on its own terms. The citations follow the usual active-matter and pattern-formation literature without obvious gaps. This work is aimed at readers already comfortable with amplitude equations and symmetry methods in active materials or mechanobiology. Someone looking for a symmetry-based organizing principle for 1D active solids will find the framework useful even if the details need tightening. It deserves peer review because the core construction is coherent and the potential application to tissue signaling is worth testing, provided the referees focus on the validity of the reduction at the COD point.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a 1D active solid model in which elasticity is reciprocally coupled to a lattice of Hopf oscillators. Amplitude equations are derived from this coupling and combined with group-theoretic analysis to classify a landscape of symmetry-protected phases; a universal compression-driven oscillation death (COD) transition is identified and proposed as a mechanism for localized signaling dampening in biological tissues.

Significance. If the amplitude-equation reduction remains valid through the COD point and the symmetry classification is free of hidden parameter dependence, the work supplies a symmetry-based taxonomy for self-organization in active solids that could unify disparate observations in mechanochemical systems. The explicit link to tissue-level signaling dampening is a concrete biophysical implication.

major comments (2)

[Amplitude equations and COD transition] The derivation of the amplitude equations (presumably in the section following the model definition) assumes that the oscillators remain in the limit-cycle regime with slowly varying amplitudes up to the COD transition. In a 1D chain under increasing compression, local amplitude can reach zero before the slow-variation assumption holds, converting the purported symmetry-protected COD into an ordinary supercritical Hopf bifurcation. An explicit center-manifold or multiple-scale validity check at the transition is required to substantiate the universality claim.
[Group-theoretic analysis] The group-theoretic classification of phases relies on the reciprocal elasticity-Hopf coupling preserving the requisite symmetries. The manuscript should demonstrate that no effective parameters are introduced by the coupling that would break the symmetry protection or render the phase diagram non-universal (cf. the claim that classification is 'purely through symmetry arguments').

minor comments (1)

[Abstract] The abstract states that the COD transition 'resolves inconsistencies in previous models' without naming the models or the inconsistencies; a single sentence identifying them would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate the revisions that will be made to strengthen the presentation of the amplitude-equation validity and the symmetry arguments.

read point-by-point responses

Referee: [Amplitude equations and COD transition] The derivation of the amplitude equations (presumably in the section following the model definition) assumes that the oscillators remain in the limit-cycle regime with slowly varying amplitudes up to the COD transition. In a 1D chain under increasing compression, local amplitude can reach zero before the slow-variation assumption holds, converting the purported symmetry-protected COD into an ordinary supercritical Hopf bifurcation. An explicit center-manifold or multiple-scale validity check at the transition is required to substantiate the universality claim.

Authors: We agree that an explicit validity check is required. In the revised manuscript we will add a center-manifold analysis together with direct numerical integration of the full oscillator-elasticity system. These checks confirm that, for the adiabatic compression rates used, the slow-amplitude assumption remains valid through the COD point and that the transition retains its symmetry-protected character rather than reducing to a standard supercritical Hopf bifurcation. revision: yes
Referee: [Group-theoretic analysis] The group-theoretic classification of phases relies on the reciprocal elasticity-Hopf coupling preserving the requisite symmetries. The manuscript should demonstrate that no effective parameters are introduced by the coupling that would break the symmetry protection or render the phase diagram non-universal (cf. the claim that classification is 'purely through symmetry arguments').

Authors: The reciprocal coupling is constructed to be invariant under the full symmetry group of the one-dimensional chain (translations and reflections). In the revised manuscript we will insert an explicit symmetry-transformation table for the coupling terms, demonstrating that no additional parameters appear that break these symmetries. Consequently the phase classification remains universal and rests solely on group-theoretic arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in symmetry and amplitude analysis

full rationale

The provided abstract and context describe a model coupling elasticity to Hopf oscillators, deriving amplitude equations, and applying group theory to classify phases and identify a COD transition. No equations, parameter fits, or self-citations are quoted that reduce any prediction or phase classification to an input by construction. The central claims rest on the reciprocal coupling framework and symmetry arguments presented as derived outputs rather than redefined inputs. Absent explicit text showing a fitted coupling renamed as a universal prediction or a self-citation chain justifying uniqueness, the derivation does not exhibit the enumerated circular patterns. This is the expected honest non-finding for a symmetry-based classification paper whose details are not shown to collapse into tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details are deferred to the full manuscript which is unavailable here.

pith-pipeline@v0.9.0 · 5621 in / 1106 out tokens · 31389 ms · 2026-05-22T19:06:43.887723+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our model reveals a rich landscape of symmetry-protected phases, identified through amplitude equations and group-theoretic analysis. We uncover a universal transition to compression-driven oscillation death (COD)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the underlying symmetry of HBR implies that the observed patterns and phases may generically arise in many natural and synthetic systems

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 4 internal anchors

[1]

W. Lim, B. Mayer, and T. Pawson, Cell signaling (Gar- land Science, 2014)

work page 2014
[2]

Gross, K

P. Gross, K. V. Kumar, and S. W. Grill, How active mechanics and regulatory biochemistry combine to form patterns in development, Annual review of biophysics46, 337 (2017)

work page 2017
[3]

Hannezo and C.-P

E. Hannezo and C.-P. Heisenberg, Mechanochemical Feedback Loops in Development and Disease, Cell 178, 12 (2019), publisher: Elsevier

work page 2019
[4]

D. B. Br¨ uckner and E. Hannezo, Tissue Active Mat- ter: Integrating Mechanics and Signaling into Dynam- ical Models, Cold Spring Harbor Perspectives in Biology , a041653 (2024), company: Cold Spring Harbor Labora- tory Press Distributor: Cold Spring Harbor Laboratory Press Institution: Cold Spring Harbor Laboratory Press Label: Cold Spring Harbor Laborat...

work page 2024
[5]

Ioratim-Uba, T

A. Ioratim-Uba, T. B. Liverpool, and S. Henkes, Mechanochemical Active Feedback Generates Conver- gence Extension in Epithelial Tissue, Physical Review Letters 131, 238301 (2023)

work page 2023
[6]

H. Lan, Q. Wang, R. Fernandez-Gonzalez, and J. J. Feng, A biomechanical model for cell polarization and interca- lation during drosophila germband extension, Physical biology 12, 056011 (2015)

work page 2015
[7]

Banerjee, K

S. Banerjee, K. J. Utuje, and M. C. Marchetti, Propa- gating stress waves during epithelial expansion, Physical review letters 114, 228101 (2015)

work page 2015
[8]

Sknepnek, I

R. Sknepnek, I. Djafer-Cherif, M. Chuai, C. Weijer, and S. Henkes, Generating active t1 transitions through mechanochemical feedback, Elife 12, e79862 (2023)

work page 2023
[9]

Dierkes, A

K. Dierkes, A. Sumi, J. Solon, and G. Salbreux, Spon- taneous oscillations of elastic contractile materials with turnover, Physical review letters 113, 148102 (2014)

work page 2014
[10]

M. F. Staddon, K. E. Cavanaugh, E. M. Munro, M. L. Gardel, and S. Banerjee, Mechanosensitive junction re- modeling promotes robust epithelial morphogenesis, Bio- physical Journal 117, 1739 (2019)

work page 2019
[11]

K. E. Cavanaugh, M. F. Staddon, S. Banerjee, and M. L. Gardel, Adaptive viscoelasticity of epithelial cell junc- tions: from models to methods, Current opinion in ge- netics & development 63, 86 (2020)

work page 2020
[12]

Etournay, M

R. Etournay, M. Popovi´ c, M. Merkel, A. Nandi, C. Blasse, B. Aigouy, H. Brandl, G. Myers, G. Salbreux, F. J¨ ulicher,et al. , Interplay of cell dynamics and epithe- lial tension during morphogenesis of the drosophila pupal wing, Elife 4, e07090 (2015)

work page 2015
[13]

N. Noll, M. Mani, I. Heemskerk, S. J. Streichan, and B. I. Shraiman, Active tension network model suggests an exotic mechanical state realized in epithelial tissues, Nature physics 13, 1221 (2017)

work page 2017
[14]

T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeo- mans, and B. Ladoux, Topological defects in epithelia govern cell death and extrusion, Nature 544, 212 (2017)

work page 2017
[15]

S. J. Streichan, M. F. Lefebvre, N. Noll, E. F. Wieschaus, and B. I. Shraiman, Global morphogenetic flow is accu- rately predicted by the spatial distribution of myosin mo- tors, Elife 7, e27454 (2018)

work page 2018
[16]

Ibrahimi and M

M. Ibrahimi and M. Merkel, Deforming polar active mat- ter in a scalar field gradient, New Journal of Physics 25, 013022 (2023)

work page 2023
[17]

Cl´ ement, B

R. Cl´ ement, B. Dehapiot, C. Collinet, T. Lecuit, and P.-F. Lenne, Viscoelastic dissipation stabilizes cell shape changes during tissue morphogenesis, Current biology 27, 3132 (2017)

work page 2017
[18]

Khalilgharibi, J

N. Khalilgharibi, J. Fouchard, N. Asadipour, R. Bar- rientos, M. Duda, A. Bonfanti, A. Yonis, A. Harris, P. Mosaffa, Y. Fujita, et al. , Stress relaxation in epithe- lial monolayers is controlled by the actomyosin cortex, Nature physics 15, 839 (2019)

work page 2019
[19]

W¨ urthner, A

L. W¨ urthner, A. Goychuk, and E. Frey, Geometry- induced patterns through mechanochemical coupling, Physical Review E 108, 014404 (2023)

work page 2023
[20]

Banerjee, T

T. Banerjee, T. Desaleux, J. Ranft, and ´E. Fodor, Hy- drodynamics of pulsating active liquids, arXiv preprint arXiv:2407.19955 (2024)

work page arXiv 2024
[21]

Mathematical modelling of calcium signalling taking into account mechanical effects

K. Kaouri, P. K. Maini, and S. J. Chapman, Mathemati- cal modelling of calcium signalling taking into account mechanical effects, arXiv preprint arXiv:1703.00540 (2017). 7

work page internal anchor Pith review Pith/arXiv arXiv 2017
[22]

Effect of curvature and normal forces on motor regulation of cilia

P. Sartori, Effect of curvature and normal forces on mo- tor regulation of cilia, arXiv preprint arXiv:1905.04138 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1905
[23]

S. A. Rautu, R. G. Morris, and M. Rao, Active mor- phodynamics of intracellular organelles in the trafficking pathway, arXiv preprint arXiv:2409.19084 (2024)

work page arXiv 2024
[24]

Dhanuka, D

A. Dhanuka, D. S. Banerjee, M. Rao, et al. , Excitability and travelling waves in renewable active matter, arXiv preprint arXiv:2503.19687 (2025)

work page arXiv 2025
[25]

De Belly, E

H. De Belly, E. K. Paluch, and K. J. Chalut, Interplay between mechanics and signalling in regulating cell fate, Nature Reviews Molecular Cell Biology 23, 465 (2022)

work page 2022
[26]

Boocock, Interplay between Mechanochemical Pat- terning and Glassy Dynamics in Cellular Monolayers, PRX Life 1, 10.1103/PRXLife.1.013001 (2023)

D. Boocock, Interplay between Mechanochemical Pat- terning and Glassy Dynamics in Cellular Monolayers, PRX Life 1, 10.1103/PRXLife.1.013001 (2023)

work page doi:10.1103/prxlife.1.013001 2023
[27]

Boocock, N

D. Boocock, N. Hino, N. Ruzickova, T. Hirashima, and E. Hannezo, Theory of mechanochemical patterning and optimal migration in cell monolayers, Nature Physics 17, 267 (2021), publisher: Nature Publishing Group

work page 2021
[28]

P´ erez-Verdugo, S

F. P´ erez-Verdugo, S. Banks, and S. Banerjee, Excitable dynamics driven by mechanical feedback in biological tissues, Communications Physics 7, 1 (2024), publisher: Nature Publishing Group

work page 2024
[29]

Parmar, L

T. Parmar, L. P. Dow, B. L. Pruitt, and M. C. Marchetti, Spontaneous and induced oscillations in confined epithe- lia, PRX Life 3, 013002 (2025)

work page 2025
[30]

Shankaran, D

H. Shankaran, D. L. Ippolito, W. B. Chrisler, H. Resat, N. Bollinger, L. K. Opresko, and H. S. Wiley, Rapid and sustained nuclear–cytoplasmic ERK oscillations induced by epidermal growth factor, Molecular Systems Biology 5, 332 (2009)

work page 2009
[31]

Moreno, L

E. Moreno, L. Valon, F. Levillayer, and R. Levayer, Competition for Space Induces Cell Elimination through Compaction-Driven ERK Downregulation, Current Biol- ogy 29, 23 (2019), publisher: Elsevier

work page 2019
[32]

N. Hino, L. Rossetti, A. Mar´ ın-Llaurad´ o, K. Aoki, X. Trepat, M. Matsuda, and T. Hirashima, Erk-mediated mechanochemical waves direct collective cell polariza- tion, Developmental cell 53, 646 (2020)

work page 2020
[33]

J. C. Barros and C. J. Marshall, Activation of either erk1/2 or erk5 map kinase pathways can lead to disrup- tion of the actin cytoskeleton, Journal of cell science118, 1663 (2005)

work page 2005
[34]

J. S. Logue, A. X. Cartagena-Rivera, M. A. Baird, M. W. Davidson, R. S. Chadwick, and C. M. Waterman, Erk regulation of actin capping and bundling by eps8 pro- motes cortex tension and leader bleb-based migration, elife 4, e08314 (2015)

work page 2015
[35]

Lavoie, J

H. Lavoie, J. Gagnon, and M. Therrien, Erk signalling: a master regulator of cell behaviour, life and fate, Nature reviews Molecular cell biology 21, 607 (2020)

work page 2020
[36]

M. C. Mendoza, E. E. Er, W. Zhang, B. A. Ballif, H. L. Elliott, G. Danuser, and J. Blenis, Erk-mapk drives lamellipodia protrusion by activating the wave2 regula- tory complex, Molecular cell 41, 661 (2011)

work page 2011
[37]

M. C. Mendoza, M. Vilela, J. E. Juarez, J. Blenis, and G. Danuser, Erk reinforces actin polymerization to power persistent edge protrusion during motility, Science signal- ing 8, ra47 (2015)

work page 2015
[38]

Tanimura and K

S. Tanimura and K. Takeda, Erk signalling as a regulator of cell motility, The Journal of Biochemistry 162, 145 (2017)

work page 2017
[39]

De Belly, A

H. De Belly, A. Stubb, A. Yanagida, C. Labouesse, P. H. Jones, E. K. Paluch, and K. J. Chalut, Membrane tension gates erk-mediated regulation of pluripotent cell fate, Cell stem cell 28, 273 (2021)

work page 2021
[40]

B. N. Kholodenko, J. F. Hancock, and W. Kolch, Sig- nalling ballet in space and time, Nature reviews Molecu- lar cell biology 11, 414 (2010)

work page 2010
[41]

Arkun and M

Y. Arkun and M. Yasemi, Dynamics and control of the erk signaling pathway: Sensitivity, bistability, and oscil- lations, PloS one 13, e0195513 (2018)

work page 2018
[42]

K. Aoki, Y. Kondo, H. Naoki, T. Hiratsuka, R. E. Itoh, and M. Matsuda, Propagating wave of erk activation ori- ents collective cell migration, Developmental cell 43, 305 (2017)

work page 2017
[43]

J. J. Tyson, Biochemical Oscillations, in Computational Cell Biology , Vol. 20, edited by C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson (Springer New York, New York, NY, 2004) pp. 230–260, series Title: Interdisci- plinary Applied Mathematics

work page 2004
[44]

I. R. Epstein and J. A. Pojman, An introduction to non- linear chemical dynamics: oscillations, waves, patterns, and chaos (Oxford university press, 1998)

work page 1998
[45]

J. J. Collins and I. N. Stewart, Coupled nonlinear os- cillators and the symmetries of animal gaits, Journal of Nonlinear Science 3, 349 (1993)

work page 1993
[46]

J. J. Collins and I. Stewart, A group-theoretic approach to rings of coupled biological oscillators, Biological Cy- bernetics 71, 95 (1994)

work page 1994
[47]

I. R. Epstein and M. Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos: An Interdisciplinary Journal of Nonlinear Science 3, 1 (1993)

work page 1993
[48]

International Centre for Theoretical Sciences, Dynamical phase transitions in Markov processes by Hugo Touchette (2019)

work page 2019
[49]

S. H. Strogatz, Nonlinear dynamics and chaos: with ap- plications to physics, biology, chemistry, and engineering (Chapman and Hall/CRC, 2024)

work page 2024
[50]

Mora and W

T. Mora and W. Bialek, Are biological systems poised at criticality?, Journal of Statistical Physics 144, 268 (2011)

work page 2011
[51]

M. A. Munoz, Colloquium: Criticality and dynamical scaling in living systems, Reviews of Modern Physics 90, 031001 (2018)

work page 2018
[52]

Ashida, Z

Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020)

work page 2020
[53]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. Irvine, and V. Vitelli, Odd elasticity, Nature Physics 16, 475 (2020)

work page 2020
[54]

Banerjee, V

D. Banerjee, V. Vitelli, F. J¨ ulicher, and P. Sur´ owka, Ac- tive viscoelasticity of odd materials, Physical Review Let- ters 126, 138001 (2021)

work page 2021
[55]

Koseska, E

A. Koseska, E. Volkov, and J. Kurths, Oscillation quench- ing mechanisms: Amplitude vs. oscillation death, Physics Reports 531, 173 (2013)

work page 2013
[56]

Fruchart, R

M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature 592, 363 (2021)

work page 2021
[57]

Davidson, Turbulence: an introduction for scientists and engineers (Oxford university press, 2015)

P. Davidson, Turbulence: an introduction for scientists and engineers (Oxford university press, 2015)

work page 2015
[58]

P. T. Nyawo and H. Touchette, A minimal model of dy- namical phase transition, Europhysics Letters116, 50009 (2017)

work page 2017
[59]

Wechselberger, Canards, Scholarpedia 2, 1356 (2007)

M. Wechselberger, Canards, Scholarpedia 2, 1356 (2007)

work page 2007
[60]

Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators

Y. Kuramoto and D. Battogtokh, Coexistence of coher- ence and incoherence in nonlocally coupled phase oscil- lators, arXiv preprint cond-mat/0210694 (2002). 8

work page internal anchor Pith review Pith/arXiv arXiv 2002
[61]

D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators, Physical review letters 93, 174102 (2004)

work page 2004
[62]

E. A. Martens, S. Thutupalli, A. Fourri` ere, and O. Hal- latschek, Chimera states in mechanical oscillator net- works, Proceedings of the National Academy of Sciences 110, 10563 (2013)

work page 2013
[63]

J. Das, A. Chaudhuri, and S. Sinha, From order to chimeras: Unraveling dynamic patterns in active fluids with nonlinear growth, arXiv preprint arXiv:2412.14729 (2024)

work page internal anchor Pith review arXiv 2024
[64]

Sindhu M. et al. unpublished data

work page
[65]

Roshan and R

N. Roshan and R. Pandit, Multiscale studies of delayed afterdepolarizations i: A comparison of two biophysically realistic mathematical models for human ventricular my- ocytes, arXiv preprint arXiv:2307.10134 (2023)

work page arXiv 2023
[66]

Tompkins, N

N. Tompkins, N. Li, C. Girabawe, M. Heymann, G. B. Er- mentrout, I. R. Epstein, and S. Fraden, Testing turing’s theory of morphogenesis in chemical cells, Proceedings of the National Academy of Sciences 111, 4397 (2014)

work page 2014
[67]

Kumar, A

M. Kumar, A. Murali, A. G. Subramaniam, R. Singh, and S. Thutupalli, Emergent dynamics due to chemo- hydrodynamic self-interactions in active polymers, Na- ture Communications 15, 4903 (2024)

work page 2024
[68]

Y. Zhai, I. Z. Kiss, and J. L. Hudson, Amplitude death through a hopf bifurcation in coupled electrochemical os- cillators: Experiments and simulations, Physical Review E 69, 026208 (2004)

work page 2004
[69]

Baconnier, D

P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Selective and collective actuation in active solids, Nature Physics 18, 1234 (2022)

work page 2022
[70]

Arora, S

P. Arora, S. Sadhukhan, S. K. Nandi, D. Bi, A. Sood, and R. Ganapathy, A shape-driven reentrant jamming tran- sition in confluent monolayers of synthetic cell-mimics, Nature Communications 15, 5645 (2024)

work page 2024
[71]

D. C. Brody, Biorthogonal quantum mechanics, Journal of Physics A: Mathematical and Theoretical 47, 035305 (2013). 9 SUPPLEMENTARY INFORMATION: OSCILLATION DEATH BY MECHANOCHEMICAL FEEDBACK I. METHODS A. Why five variables? Our model solves the coupled dynamics of five variables: M, E, D, L0, R. L0 and R are the parameters of the spring and cannot be remo...

work page 2013
[72]

They are the unit of length and time in our model

a = 1 and τR = 1 for all simulations. They are the unit of length and time in our model

work page
[73]

A single seed value is shown

For Fig 3A-C, we have simulated a model with Lx = 200a and tmax = 5 × 104τR. A single seed value is shown

work page
[74]

Lx = 1000a, tmax = 105τR

For Fig 4A-B, we have chosen α = β = √µ, where µ ∈ [1e − 2, 9]. Lx = 1000a, tmax = 105τR. The distributions were obtained from datapoints from t = 9.9 × 104 − 105 and averaged over 10 random initial conditions. All together, 103 × 103 × 10 = 107 samples were used to compute the distributions

work page
[75]

S5, S6, we have chosen α, β = {0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 2.5, 3}

For Fig 4C and Fig. S5, S6, we have chosen α, β = {0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 2.5, 3}. For each of the 100 pairs of α, β values and 9 pairs of τD, τL values, we used 20 random initial conditions

work page
[76]

4D, check section I.E

For details of Fig. 4D, check section I.E

work page
[77]

10 different replicates were used

For Fig.4E, we chose µ = µc ± δµ, where µc = 0.5, 3.24, and 5.60 and δµ ∈ [10−6, 10−1]. 10 different replicates were used. FIG. S2. Numerical estimate of LOD at short (left) and long timescales (right). The white lines show our estimate. The color map indicates the value of ∂xR. D. Numerical estimation of LOD We observed that E does not change appreciably...

work page

[1] [1]

W. Lim, B. Mayer, and T. Pawson, Cell signaling (Gar- land Science, 2014)

work page 2014

[2] [2]

Gross, K

P. Gross, K. V. Kumar, and S. W. Grill, How active mechanics and regulatory biochemistry combine to form patterns in development, Annual review of biophysics46, 337 (2017)

work page 2017

[3] [3]

Hannezo and C.-P

E. Hannezo and C.-P. Heisenberg, Mechanochemical Feedback Loops in Development and Disease, Cell 178, 12 (2019), publisher: Elsevier

work page 2019

[4] [4]

D. B. Br¨ uckner and E. Hannezo, Tissue Active Mat- ter: Integrating Mechanics and Signaling into Dynam- ical Models, Cold Spring Harbor Perspectives in Biology , a041653 (2024), company: Cold Spring Harbor Labora- tory Press Distributor: Cold Spring Harbor Laboratory Press Institution: Cold Spring Harbor Laboratory Press Label: Cold Spring Harbor Laborat...

work page 2024

[5] [5]

Ioratim-Uba, T

A. Ioratim-Uba, T. B. Liverpool, and S. Henkes, Mechanochemical Active Feedback Generates Conver- gence Extension in Epithelial Tissue, Physical Review Letters 131, 238301 (2023)

work page 2023

[6] [6]

H. Lan, Q. Wang, R. Fernandez-Gonzalez, and J. J. Feng, A biomechanical model for cell polarization and interca- lation during drosophila germband extension, Physical biology 12, 056011 (2015)

work page 2015

[7] [7]

Banerjee, K

S. Banerjee, K. J. Utuje, and M. C. Marchetti, Propa- gating stress waves during epithelial expansion, Physical review letters 114, 228101 (2015)

work page 2015

[8] [8]

Sknepnek, I

R. Sknepnek, I. Djafer-Cherif, M. Chuai, C. Weijer, and S. Henkes, Generating active t1 transitions through mechanochemical feedback, Elife 12, e79862 (2023)

work page 2023

[9] [9]

Dierkes, A

K. Dierkes, A. Sumi, J. Solon, and G. Salbreux, Spon- taneous oscillations of elastic contractile materials with turnover, Physical review letters 113, 148102 (2014)

work page 2014

[10] [10]

M. F. Staddon, K. E. Cavanaugh, E. M. Munro, M. L. Gardel, and S. Banerjee, Mechanosensitive junction re- modeling promotes robust epithelial morphogenesis, Bio- physical Journal 117, 1739 (2019)

work page 2019

[11] [11]

K. E. Cavanaugh, M. F. Staddon, S. Banerjee, and M. L. Gardel, Adaptive viscoelasticity of epithelial cell junc- tions: from models to methods, Current opinion in ge- netics & development 63, 86 (2020)

work page 2020

[12] [12]

Etournay, M

R. Etournay, M. Popovi´ c, M. Merkel, A. Nandi, C. Blasse, B. Aigouy, H. Brandl, G. Myers, G. Salbreux, F. J¨ ulicher,et al. , Interplay of cell dynamics and epithe- lial tension during morphogenesis of the drosophila pupal wing, Elife 4, e07090 (2015)

work page 2015

[13] [13]

N. Noll, M. Mani, I. Heemskerk, S. J. Streichan, and B. I. Shraiman, Active tension network model suggests an exotic mechanical state realized in epithelial tissues, Nature physics 13, 1221 (2017)

work page 2017

[14] [14]

T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeo- mans, and B. Ladoux, Topological defects in epithelia govern cell death and extrusion, Nature 544, 212 (2017)

work page 2017

[15] [15]

S. J. Streichan, M. F. Lefebvre, N. Noll, E. F. Wieschaus, and B. I. Shraiman, Global morphogenetic flow is accu- rately predicted by the spatial distribution of myosin mo- tors, Elife 7, e27454 (2018)

work page 2018

[16] [16]

Ibrahimi and M

M. Ibrahimi and M. Merkel, Deforming polar active mat- ter in a scalar field gradient, New Journal of Physics 25, 013022 (2023)

work page 2023

[17] [17]

Cl´ ement, B

R. Cl´ ement, B. Dehapiot, C. Collinet, T. Lecuit, and P.-F. Lenne, Viscoelastic dissipation stabilizes cell shape changes during tissue morphogenesis, Current biology 27, 3132 (2017)

work page 2017

[18] [18]

Khalilgharibi, J

N. Khalilgharibi, J. Fouchard, N. Asadipour, R. Bar- rientos, M. Duda, A. Bonfanti, A. Yonis, A. Harris, P. Mosaffa, Y. Fujita, et al. , Stress relaxation in epithe- lial monolayers is controlled by the actomyosin cortex, Nature physics 15, 839 (2019)

work page 2019

[19] [19]

W¨ urthner, A

L. W¨ urthner, A. Goychuk, and E. Frey, Geometry- induced patterns through mechanochemical coupling, Physical Review E 108, 014404 (2023)

work page 2023

[20] [20]

Banerjee, T

T. Banerjee, T. Desaleux, J. Ranft, and ´E. Fodor, Hy- drodynamics of pulsating active liquids, arXiv preprint arXiv:2407.19955 (2024)

work page arXiv 2024

[21] [21]

Mathematical modelling of calcium signalling taking into account mechanical effects

K. Kaouri, P. K. Maini, and S. J. Chapman, Mathemati- cal modelling of calcium signalling taking into account mechanical effects, arXiv preprint arXiv:1703.00540 (2017). 7

work page internal anchor Pith review Pith/arXiv arXiv 2017

[22] [22]

Effect of curvature and normal forces on motor regulation of cilia

P. Sartori, Effect of curvature and normal forces on mo- tor regulation of cilia, arXiv preprint arXiv:1905.04138 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1905

[23] [23]

S. A. Rautu, R. G. Morris, and M. Rao, Active mor- phodynamics of intracellular organelles in the trafficking pathway, arXiv preprint arXiv:2409.19084 (2024)

work page arXiv 2024

[24] [24]

Dhanuka, D

A. Dhanuka, D. S. Banerjee, M. Rao, et al. , Excitability and travelling waves in renewable active matter, arXiv preprint arXiv:2503.19687 (2025)

work page arXiv 2025

[25] [25]

De Belly, E

H. De Belly, E. K. Paluch, and K. J. Chalut, Interplay between mechanics and signalling in regulating cell fate, Nature Reviews Molecular Cell Biology 23, 465 (2022)

work page 2022

[26] [26]

Boocock, Interplay between Mechanochemical Pat- terning and Glassy Dynamics in Cellular Monolayers, PRX Life 1, 10.1103/PRXLife.1.013001 (2023)

D. Boocock, Interplay between Mechanochemical Pat- terning and Glassy Dynamics in Cellular Monolayers, PRX Life 1, 10.1103/PRXLife.1.013001 (2023)

work page doi:10.1103/prxlife.1.013001 2023

[27] [27]

Boocock, N

D. Boocock, N. Hino, N. Ruzickova, T. Hirashima, and E. Hannezo, Theory of mechanochemical patterning and optimal migration in cell monolayers, Nature Physics 17, 267 (2021), publisher: Nature Publishing Group

work page 2021

[28] [28]

P´ erez-Verdugo, S

F. P´ erez-Verdugo, S. Banks, and S. Banerjee, Excitable dynamics driven by mechanical feedback in biological tissues, Communications Physics 7, 1 (2024), publisher: Nature Publishing Group

work page 2024

[29] [29]

Parmar, L

T. Parmar, L. P. Dow, B. L. Pruitt, and M. C. Marchetti, Spontaneous and induced oscillations in confined epithe- lia, PRX Life 3, 013002 (2025)

work page 2025

[30] [30]

Shankaran, D

H. Shankaran, D. L. Ippolito, W. B. Chrisler, H. Resat, N. Bollinger, L. K. Opresko, and H. S. Wiley, Rapid and sustained nuclear–cytoplasmic ERK oscillations induced by epidermal growth factor, Molecular Systems Biology 5, 332 (2009)

work page 2009

[31] [31]

Moreno, L

E. Moreno, L. Valon, F. Levillayer, and R. Levayer, Competition for Space Induces Cell Elimination through Compaction-Driven ERK Downregulation, Current Biol- ogy 29, 23 (2019), publisher: Elsevier

work page 2019

[32] [32]

N. Hino, L. Rossetti, A. Mar´ ın-Llaurad´ o, K. Aoki, X. Trepat, M. Matsuda, and T. Hirashima, Erk-mediated mechanochemical waves direct collective cell polariza- tion, Developmental cell 53, 646 (2020)

work page 2020

[33] [33]

J. C. Barros and C. J. Marshall, Activation of either erk1/2 or erk5 map kinase pathways can lead to disrup- tion of the actin cytoskeleton, Journal of cell science118, 1663 (2005)

work page 2005

[34] [34]

J. S. Logue, A. X. Cartagena-Rivera, M. A. Baird, M. W. Davidson, R. S. Chadwick, and C. M. Waterman, Erk regulation of actin capping and bundling by eps8 pro- motes cortex tension and leader bleb-based migration, elife 4, e08314 (2015)

work page 2015

[35] [35]

Lavoie, J

H. Lavoie, J. Gagnon, and M. Therrien, Erk signalling: a master regulator of cell behaviour, life and fate, Nature reviews Molecular cell biology 21, 607 (2020)

work page 2020

[36] [36]

M. C. Mendoza, E. E. Er, W. Zhang, B. A. Ballif, H. L. Elliott, G. Danuser, and J. Blenis, Erk-mapk drives lamellipodia protrusion by activating the wave2 regula- tory complex, Molecular cell 41, 661 (2011)

work page 2011

[37] [37]

M. C. Mendoza, M. Vilela, J. E. Juarez, J. Blenis, and G. Danuser, Erk reinforces actin polymerization to power persistent edge protrusion during motility, Science signal- ing 8, ra47 (2015)

work page 2015

[38] [38]

Tanimura and K

S. Tanimura and K. Takeda, Erk signalling as a regulator of cell motility, The Journal of Biochemistry 162, 145 (2017)

work page 2017

[39] [39]

De Belly, A

H. De Belly, A. Stubb, A. Yanagida, C. Labouesse, P. H. Jones, E. K. Paluch, and K. J. Chalut, Membrane tension gates erk-mediated regulation of pluripotent cell fate, Cell stem cell 28, 273 (2021)

work page 2021

[40] [40]

B. N. Kholodenko, J. F. Hancock, and W. Kolch, Sig- nalling ballet in space and time, Nature reviews Molecu- lar cell biology 11, 414 (2010)

work page 2010

[41] [41]

Arkun and M

Y. Arkun and M. Yasemi, Dynamics and control of the erk signaling pathway: Sensitivity, bistability, and oscil- lations, PloS one 13, e0195513 (2018)

work page 2018

[42] [42]

K. Aoki, Y. Kondo, H. Naoki, T. Hiratsuka, R. E. Itoh, and M. Matsuda, Propagating wave of erk activation ori- ents collective cell migration, Developmental cell 43, 305 (2017)

work page 2017

[43] [43]

J. J. Tyson, Biochemical Oscillations, in Computational Cell Biology , Vol. 20, edited by C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson (Springer New York, New York, NY, 2004) pp. 230–260, series Title: Interdisci- plinary Applied Mathematics

work page 2004

[44] [44]

I. R. Epstein and J. A. Pojman, An introduction to non- linear chemical dynamics: oscillations, waves, patterns, and chaos (Oxford university press, 1998)

work page 1998

[45] [45]

J. J. Collins and I. N. Stewart, Coupled nonlinear os- cillators and the symmetries of animal gaits, Journal of Nonlinear Science 3, 349 (1993)

work page 1993

[46] [46]

J. J. Collins and I. Stewart, A group-theoretic approach to rings of coupled biological oscillators, Biological Cy- bernetics 71, 95 (1994)

work page 1994

[47] [47]

I. R. Epstein and M. Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos: An Interdisciplinary Journal of Nonlinear Science 3, 1 (1993)

work page 1993

[48] [48]

International Centre for Theoretical Sciences, Dynamical phase transitions in Markov processes by Hugo Touchette (2019)

work page 2019

[49] [49]

S. H. Strogatz, Nonlinear dynamics and chaos: with ap- plications to physics, biology, chemistry, and engineering (Chapman and Hall/CRC, 2024)

work page 2024

[50] [50]

Mora and W

T. Mora and W. Bialek, Are biological systems poised at criticality?, Journal of Statistical Physics 144, 268 (2011)

work page 2011

[51] [51]

M. A. Munoz, Colloquium: Criticality and dynamical scaling in living systems, Reviews of Modern Physics 90, 031001 (2018)

work page 2018

[52] [52]

Ashida, Z

Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020)

work page 2020

[53] [53]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. Irvine, and V. Vitelli, Odd elasticity, Nature Physics 16, 475 (2020)

work page 2020

[54] [54]

Banerjee, V

D. Banerjee, V. Vitelli, F. J¨ ulicher, and P. Sur´ owka, Ac- tive viscoelasticity of odd materials, Physical Review Let- ters 126, 138001 (2021)

work page 2021

[55] [55]

Koseska, E

A. Koseska, E. Volkov, and J. Kurths, Oscillation quench- ing mechanisms: Amplitude vs. oscillation death, Physics Reports 531, 173 (2013)

work page 2013

[56] [56]

Fruchart, R

M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature 592, 363 (2021)

work page 2021

[57] [57]

Davidson, Turbulence: an introduction for scientists and engineers (Oxford university press, 2015)

P. Davidson, Turbulence: an introduction for scientists and engineers (Oxford university press, 2015)

work page 2015

[58] [58]

P. T. Nyawo and H. Touchette, A minimal model of dy- namical phase transition, Europhysics Letters116, 50009 (2017)

work page 2017

[59] [59]

Wechselberger, Canards, Scholarpedia 2, 1356 (2007)

M. Wechselberger, Canards, Scholarpedia 2, 1356 (2007)

work page 2007

[60] [60]

Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators

Y. Kuramoto and D. Battogtokh, Coexistence of coher- ence and incoherence in nonlocally coupled phase oscil- lators, arXiv preprint cond-mat/0210694 (2002). 8

work page internal anchor Pith review Pith/arXiv arXiv 2002

[61] [61]

D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators, Physical review letters 93, 174102 (2004)

work page 2004

[62] [62]

E. A. Martens, S. Thutupalli, A. Fourri` ere, and O. Hal- latschek, Chimera states in mechanical oscillator net- works, Proceedings of the National Academy of Sciences 110, 10563 (2013)

work page 2013

[63] [63]

J. Das, A. Chaudhuri, and S. Sinha, From order to chimeras: Unraveling dynamic patterns in active fluids with nonlinear growth, arXiv preprint arXiv:2412.14729 (2024)

work page internal anchor Pith review arXiv 2024

[64] [64]

Sindhu M. et al. unpublished data

work page

[65] [65]

Roshan and R

N. Roshan and R. Pandit, Multiscale studies of delayed afterdepolarizations i: A comparison of two biophysically realistic mathematical models for human ventricular my- ocytes, arXiv preprint arXiv:2307.10134 (2023)

work page arXiv 2023

[66] [66]

Tompkins, N

N. Tompkins, N. Li, C. Girabawe, M. Heymann, G. B. Er- mentrout, I. R. Epstein, and S. Fraden, Testing turing’s theory of morphogenesis in chemical cells, Proceedings of the National Academy of Sciences 111, 4397 (2014)

work page 2014

[67] [67]

Kumar, A

M. Kumar, A. Murali, A. G. Subramaniam, R. Singh, and S. Thutupalli, Emergent dynamics due to chemo- hydrodynamic self-interactions in active polymers, Na- ture Communications 15, 4903 (2024)

work page 2024

[68] [68]

Y. Zhai, I. Z. Kiss, and J. L. Hudson, Amplitude death through a hopf bifurcation in coupled electrochemical os- cillators: Experiments and simulations, Physical Review E 69, 026208 (2004)

work page 2004

[69] [69]

Baconnier, D

P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Selective and collective actuation in active solids, Nature Physics 18, 1234 (2022)

work page 2022

[70] [70]

Arora, S

P. Arora, S. Sadhukhan, S. K. Nandi, D. Bi, A. Sood, and R. Ganapathy, A shape-driven reentrant jamming tran- sition in confluent monolayers of synthetic cell-mimics, Nature Communications 15, 5645 (2024)

work page 2024

[71] [71]

D. C. Brody, Biorthogonal quantum mechanics, Journal of Physics A: Mathematical and Theoretical 47, 035305 (2013). 9 SUPPLEMENTARY INFORMATION: OSCILLATION DEATH BY MECHANOCHEMICAL FEEDBACK I. METHODS A. Why five variables? Our model solves the coupled dynamics of five variables: M, E, D, L0, R. L0 and R are the parameters of the spring and cannot be remo...

work page 2013

[72] [72]

They are the unit of length and time in our model

a = 1 and τR = 1 for all simulations. They are the unit of length and time in our model

work page

[73] [73]

A single seed value is shown

For Fig 3A-C, we have simulated a model with Lx = 200a and tmax = 5 × 104τR. A single seed value is shown

work page

[74] [74]

Lx = 1000a, tmax = 105τR

For Fig 4A-B, we have chosen α = β = √µ, where µ ∈ [1e − 2, 9]. Lx = 1000a, tmax = 105τR. The distributions were obtained from datapoints from t = 9.9 × 104 − 105 and averaged over 10 random initial conditions. All together, 103 × 103 × 10 = 107 samples were used to compute the distributions

work page

[75] [75]

S5, S6, we have chosen α, β = {0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 2.5, 3}

For Fig 4C and Fig. S5, S6, we have chosen α, β = {0.1, 0.2, 0.3, 0.5, 0.8, 1, 1.5, 2, 2.5, 3}. For each of the 100 pairs of α, β values and 9 pairs of τD, τL values, we used 20 random initial conditions

work page

[76] [76]

4D, check section I.E

For details of Fig. 4D, check section I.E

work page

[77] [77]

10 different replicates were used

For Fig.4E, we chose µ = µc ± δµ, where µc = 0.5, 3.24, and 5.60 and δµ ∈ [10−6, 10−1]. 10 different replicates were used. FIG. S2. Numerical estimate of LOD at short (left) and long timescales (right). The white lines show our estimate. The color map indicates the value of ∂xR. D. Numerical estimation of LOD We observed that E does not change appreciably...

work page