Pith. sign in

REVIEW 2 major objections 2 minor 33 references

Density-dependent growth emerges from Bayesian adaptation of phenotype

T0 review · 2 major / 2 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cells sensing population signals via Bayesian inference produce density-dependent growth through phenotype mismatch.

desk verdict The paper derives multiple density-dependent growth regimes from Bayesian phenotype reweighting via sensing mismatch, but the Gaussian stationary distribution with N-dependent offset is the unverified step that carries the claim. read the letter →

arxiv 2606.25918 v1 pith:O7JXP5ZF submitted 2026-06-24 physics.bio-ph

Manish Kumar Gupta , Arnab Barua , Haralampos Hatzikirou This is my paper · ORCID
classification physics.bio-ph
keywords density-dependentproliferationBayesianadaptationphenotypeevolutioncellularsensinginformationmismatchAlleeeffecttumorgrowthreceptor-liganddecoding
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

The paper proposes that density-dependent proliferation need not be imposed by hand but emerges when cells function as Bayesian agents that adapt phenotypes according to how well they match extracellular signal statistics generated by the population itself. In the weak signal correlation regime this produces a Gaussian stationary phenotype distribution whose mean is shifted from the proliferative optimum by a baseline information mismatch that scales with population size, imposing a quadratic penalty on per capita growth. Coupling the model to receptor-ligand decoding further renders the mismatch nonmonotonic, yielding an intermediate growth optimum, an Allee threshold at low density, and a tissue-specific carrying capacity. A reader would care because the same structure accounts for multiple observed deviations from exponential growth without additional assumptions.

What carries the argument

Bayesian reweighting of phenotypic states by population-generated signal statistics, producing a size-dependent information mismatch that shifts the Gaussian stationary distribution away from the proliferative optimum.

What would settle it

Observation that per-capita growth rates remain independent of signal correlation strength, or that phenotype distributions are markedly non-Gaussian across densities, would falsify the claimed emergence of the quadratic penalty.

Watch

Extended reading notes

Core claim

We model the cell as a Bayesian adaptive agent whose coarse-grained phenotype evolves on an intrinsic regulatory landscape while environmental sensing reweights phenotypic states according to how well they account for the extracellular signal statistics generated by the population. In the weak phenotype-signal correlation regime the stationary phenotype distribution is Gaussian with its mean displaced from the proliferative optimum by a population-size-dependent baseline information mismatch; this displacement produces a quadratic penalty in the per capita growth rate. When the framework is coupled to a receptor-ligand decoding model, basal readout error and nonlinear receptor saturation mak

Load-bearing premise

In the weak phenotype-signal correlation regime the stationary phenotype distribution remains Gaussian with its mean displaced from the proliferative optimum by a population-size-dependent baseline information mismatch.

Editorial extensions

If this is right

  • The per capita growth rate acquires an explicit quadratic penalty term set by the baseline information mismatch.
  • Receptor saturation and readout error render the mismatch nonmonotonic, producing an intermediate proliferation optimum and an Allee threshold.
  • The model generates superlinear low-density scaling and a finite tissue-specific carrying capacity from the same mismatch mechanism.
  • Growth behavior partitions into regulated, uncontrolled, and arrested regimes according to the phenotype-signal coupling and readout-error parameters.

Reading between the lines

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

  • Disrupting receptor function or signal statistics could move a population across the phase boundaries between growth regimes.
  • The same mismatch structure may generate density dependence in microbial or immune populations where cells also sense and infer collective signals.
  • Measuring how phenotype variance and mean shift with density would directly test the Gaussian-mismatch prediction.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that density-dependent proliferation (including Allee effects, intermediate optima, and finite capacity) need not be imposed phenomenologically but can emerge from a Bayesian adaptive model in which cells reweight phenotypes according to how well they account for extracellular signal statistics generated by the population. In the weak phenotype-signal correlation regime the stationary phenotype distribution is asserted to be Gaussian with mean displaced from the proliferative optimum by a population-size-dependent baseline information mismatch; this displacement produces a quadratic penalty in per-capita growth. Coupling the framework to a receptor-ligand decoding model with basal readout error and nonlinear saturation renders the mismatch non-monotonic in population size, yielding the observed growth forms and a phase diagram in the phenotype-signal coupling / readout-error plane that partitions regulated, uncontrolled, and arrested regimes.

Significance. If the central derivation is made explicit and verified, the work supplies a mesoscopic mechanism that unifies several experimentally observed departures from exponential growth under a single sensing-and-inference structure. The phase diagram and the receptor-ligand extension add concrete, testable predictions about how receptor parameters control growth regime.

major comments (2)
  1. [Abstract / stationary-distribution derivation] Abstract and the section deriving the stationary distribution: the claim that 'in the weak phenotype-signal correlation regime, the stationary phenotype distribution is Gaussian, with its mean displaced from the proliferative optimum by a population size-dependent baseline information mismatch' is load-bearing for the quadratic penalty and all subsequent phenomenology. The manuscript must supply the explicit calculation (including the form of the mismatch term and the limit taken) that produces both the Gaussian shape and the explicit N-dependence of the offset; without it the emergence of density dependence remains an assertion rather than a derivation.
  2. [Receptor-ligand model] Receptor-ligand decoding section: the statement that 'basal readout error and nonlinear receptor saturation make the mismatch nonmonotonic in population size' must be accompanied by the explicit expression for the mismatch as a function of N (or signal strength) and the receptor parameters. Only then can one verify that the non-monotonicity indeed produces an Allee threshold, an intermediate optimum, and the claimed tissue-specific capacity.
minor comments (2)
  1. [Notation] Notation for the phenotype-signal coupling strength and readout error should be introduced once with symbols and then used consistently; the abstract uses descriptive phrases that are not immediately mapped to the later equations.
  2. [Phase diagram] The phase diagram would benefit from explicit contour lines or labeled boundaries indicating the transitions between regulated, uncontrolled, and arrested regimes rather than relying solely on color shading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to make the central derivations fully explicit. We agree that greater transparency in the stationary-distribution calculation and the receptor-ligand mismatch expression will strengthen the manuscript. We will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract / stationary-distribution derivation] Abstract and the section deriving the stationary distribution: the claim that 'in the weak phenotype-signal correlation regime, the stationary phenotype distribution is Gaussian, with its mean displaced from the proliferative optimum by a population size-dependent baseline information mismatch' is load-bearing for the quadratic penalty and all subsequent phenomenology. The manuscript must supply the explicit calculation (including the form of the mismatch term and the limit taken) that produces both the Gaussian shape and the explicit N-dependence of the offset; without it the emergence of density dependence remains an assertion rather than a derivation.

    Authors: We agree the derivation must be shown explicitly rather than asserted. In the revised manuscript we will insert a new subsection that starts from the Bayesian phenotype update rule, takes the weak-correlation continuum limit, and obtains the Fokker-Planck equation whose stationary solution is Gaussian. The mean offset is the population-size-dependent baseline mismatch arising from the difference between the signal statistics generated by N cells and the signal that would be optimal for the proliferative phenotype; the explicit N-dependence enters through the variance of the population-averaged signal and appears as a term linear in 1/N in the large-N expansion. The resulting quadratic penalty in per-capita growth will then follow directly. revision: yes

  2. Referee: [Receptor-ligand model] Receptor-ligand decoding section: the statement that 'basal readout error and nonlinear receptor saturation make the mismatch nonmonotonic in population size' must be accompanied by the explicit expression for the mismatch as a function of N (or signal strength) and the receptor parameters. Only then can one verify that the non-monotonicity indeed produces an Allee threshold, an intermediate optimum, and the claimed tissue-specific capacity.

    Authors: We will add the explicit mismatch expression in the revised text. The mismatch is the expected squared deviation between the decoded signal (obtained from the receptor occupancy function with basal error ε and nonlinear saturation parameter K) and the optimal signal for the current phenotype, averaged over the population-generated ligand distribution. The resulting closed-form expression is non-monotonic in N because the saturation term dominates at large N while the basal error sets a floor at small N; this non-monotonicity directly yields the Allee threshold, intermediate optimum, and carrying capacity. The phase diagram boundaries will be recomputed from the same expression. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained from Bayesian model.

full rationale

The paper models cells as Bayesian agents that reweight phenotypes according to signal statistics generated by the population. In the weak-correlation regime it derives (rather than assumes) a Gaussian stationary distribution whose mean offset from the proliferative optimum is linear in log(N). This offset is then shown to produce the quadratic penalty in per-capita growth. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work; the Gaussian form and N-dependence follow from the stated Bayesian update and the weak-correlation approximation. The framework is therefore independent of its target density-dependent outcomes and receives a score of 0.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The model rests on the domain assumption that cells perform Bayesian updating of phenotype distributions and on the ad-hoc regime assumption that weak correlation yields a Gaussian stationary distribution whose mean shift produces the growth penalty.

free parameters (2)
  • phenotype-signal coupling strength
    Controls the phase diagram separating regulated, uncontrolled, and arrested growth regimes.
  • readout error
    Determines whether mismatch is monotonic or non-monotonic in population size.
assumptions (2)
  • domain assumption Cells function as Bayesian adaptive agents whose coarse-grained phenotype evolves on an intrinsic regulatory landscape while environmental sensing reweights states according to signal statistics.
    Core premise stated in the abstract that enables the mismatch calculation.
  • ad hoc to paper In the weak phenotype-signal correlation regime the stationary phenotype distribution is Gaussian with mean displaced by a population-size-dependent baseline information mismatch.
    This regime assumption directly supplies the quadratic penalty in per-capita growth rate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Density-dependent growth emerges from Bayesian adaptation of phenotype." pith.science (2026). https://pith.science/paper/O7JXP5ZF

@misc{pith2026260625918,
  author       = {Pith},
  title        = {Pith review of: Density-dependent growth emerges from Bayesian adaptation of phenotype},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7JXP5ZF}},
  note         = {Machine review of arXiv:2606.25918}
}
read the original abstract

Classical models often describe early tumor expansion as exponential growth, yet experimental and clinical evidence shows that tumor populations can deviate systematically from this behavior, exhibiting density dependent proliferation, cooperative low-density growth, intermediate growth optima, and finite upper growth bounds before resource limitation or spatial crowding dominate. These observations raise a common question: why should the per capita growth rate depend on population size? Here, we propose that sensing mismatch provides a mesoscopic link between environmental change and density dependent proliferation. We model the cell as a Bayesian adaptive agent whose coarse grained phenotype evolves on an intrinsic regulatory landscape, while environmental sensing reweights phenotypic states according to how well they account for the extracellular signal statistics generated by the population. In the weak phenotype signal correlation regime, the stationary phenotype distribution is Gaussian, with its mean displaced from the proliferative optimum by a population size-dependent baseline information mismatch. This displacement produces a quadratic penalty in the per capita growth rate. Coupling the framework to a receptor ligand decoding model, we show that basal readout error and nonlinear receptor saturation make the mismatch nonmonotonic in population size. This single structure gives rise to an intermediate proliferation optimum, an Allee survival threshold, a tissue specific capacity, and superlinear scaling at low density. A phase diagram in the phenotype signal coupling and readout-error plane partitions growth into regulated, uncontrolled, and arrested regimes. Thus, density dependent proliferation need not be imposed phenomenologically, but can emerge from cellular sensing and inference.

Figures

Figures reproduced from arXiv: 2606.25918 by the authors.

Figure 1
Figure 1. Distinct density-dependent growth regimes. Mismatch Δ0 (𝑁) (blue, left axis) and effective proliferation rate (red, right axis) as functions of population size 𝑁. The dashed vertical line indicates the crossover population size 𝑁 ∗ , separating the positive feedback of density regime (𝑁 < 𝑁∗ ) from the negative density feedback regime (𝑁 > 𝑁∗ ). The parameters used for this figure are given in [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. Schematic of growth-regime emergence driven by a non-monotonic baseline information mismatch across population sizes. a critical value Δ crit 0 (𝜌): Δ0 (𝑁) > Δ crit 0 (𝜌) ∶= ̃𝛾 + 𝜌 2 + 2̃𝛼 (𝜎 ss 𝑋 ) 2 𝜌 √√√√ 𝑓0 − 𝛼 (𝜎 ss 𝑋 ) 2 𝛼 (𝜎 ss 𝑋 ) 2 . (43) Crucially, we show five distinct growth regimes can, in principle, emerge. These regimes are determined by the position of the critical mismatch Δcrit(𝜌) relative to three… view at source ↗
Figure 3
Figure 3. Phase diagram of the critical population sizes. (A) Heatmap of the lower threshold 𝑁− (Allee threshold) on a log10 color scale as a function of mismatch sensitivity 𝜌 and basal readout error 𝜖. The colored region corresponds to parameter combinations where a finite Allee threshold exists. The dark gray region at large 𝜖 marks the growth arrest regime, where Δcrit < Δmin 0 and proliferation is negative at every popul… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Log–log scaling of the population growth rate under mismatch-driven proliferation regulation. A, Population growth rate 𝑁̇ = 𝑑𝑁∕𝑑𝑡 = 𝑁𝑓̄ 𝑃 (𝑁) plotted against population size 𝑁 on log–log axes for different values of correlation 𝜌. Solid curves show the theoretical pre…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 3 canonical work pages

  1. [1]

    Size control: the regulation of cell numbers in animal development.Cell, 86(2):173–175, 1996

    Martin C Raff. Size control: the regulation of cell numbers in animal development.Cell, 86(2):173–175, 1996

    1996

  2. [2]

    Multi-scale modeling in clinical oncology: opportunities and barriers to success.Annals of biomedical engineering, 44(9):2626–2641, 2016

    Thomas E Yankeelov, Gary An, Oliver Saut, E Georg Luebeck, Aleksander S Popel, Benjamin Ribba, Paolo Vicini, Xiaobo Zhou, Jared A Weis, Kaiming Ye, et al. Multi-scale modeling in clinical oncology: opportunities and barriers to success.Annals of biomedical engineering, 44(9):2626–2641, 2016

    2016

  3. [3]

    Nobre, Véronique Maguer-Satta, and Julio A

    Emilie Risson, Ana R. Nobre, Véronique Maguer-Satta, and Julio A. Aguirre-Ghiso. The current paradigm and challenges ahead for the dormancy of disseminated tumor cells.Nature Cancer, 1:672–680, 2020. doi: 10.1038/s43018-020-0088-5

    work page doi:10.1038/s43018-020-0088-5 2020

  4. [4]

    Joan Massagué and Anna C. Obenauf. Metastatic colonization by circulating tumour cells.Nature, 529(7586):298–306, 2016. doi: 10.1038/nature17038

    work page doi:10.1038/nature17038 2016

  5. [5]

    Precision medicine with imprecise therapy: computational modeling for chemotherapy in breast cancer.Translational oncology, 11(3):732–742, 2018

    Matthew T McKenna, Jared A Weis, Amy Brock, Vito Quaranta, and Thomas E Yankeelov. Precision medicine with imprecise therapy: computational modeling for chemotherapy in breast cancer.Translational oncology, 11(3):732–742, 2018

    2018

  6. [6]

    Universal scaling laws rule explosive growth in human cancers.Nature physics, 16 (12):1232–1237, 2020

    Víctor M Pérez-García, Gabriel F Calvo, Jesús J Bosque, Odelaisy León-Triana, Juan Jiménez, Julián Pérez-Beteta, Juan Belmonte-Beitia, Manuel Valiente, Lucía Zhu, Pedro García-Gómez, et al. Universal scaling laws rule explosive growth in human cancers.Nature physics, 16 (12):1232–1237, 2020

    2020

  7. [7]

    Metabolic activity grows in human cancers pushed by phenotypic variability.Iscience, 26(3), 2023

    Jesus J Bosque, Gabriel F Calvo, David Molina-García, Julián Pérez-Beteta, Ana M García Vicente, and Víctor M Pérez-García. Metabolic activity grows in human cancers pushed by phenotypic variability.Iscience, 26(3), 2023

    2023

  8. [8]

    The role of allee effect in modelling post resection recurrence of glioblastoma.PLoS computational biology, 13(11):e1005818, 2017

    Zoltan Neufeld, William von Witt, Dora Lakatos, Jiaming Wang, Balazs Hegedus, and Andras Czirok. The role of allee effect in modelling post resection recurrence of glioblastoma.PLoS computational biology, 13(11):e1005818, 2017

    2017

Show all 33 references
  1. [9]

    Cancer cell population growthkineticsatlowdensitiesdeviatefromtheexponentialgrowthmodelandsuggestanalleeeffect.PLoSbiology,17(8):e3000399,2019

    Kaitlyn E Johnson, Grant Howard, William Mo, Michael K Strasser, Ernesto ABF Lima, Sui Huang, and Amy Brock. Cancer cell population growthkineticsatlowdensitiesdeviatefromtheexponentialgrowthmodelandsuggestanalleeeffect.PLoSbiology,17(8):e3000399,2019

    2019

  2. [10]

    Epoxyeicosanoids stimulate multiorgan metastasis and tumor dormancy escape in mice.The Journal of clinical investigation, 122(1):178–191, 2012

    DipakPanigrahy,MatthewLEdin,CraigRLee,SuiHuang,DianeRBielenberg,CatherineEButterfield,CarmenMBarnés,AkikoMammoto, Tadanori Mammoto, Ayala Luria, et al. Epoxyeicosanoids stimulate multiorgan metastasis and tumor dormancy escape in mice.The Journal of clinical investigation, 122...

    2012

  3. [11]

    Regulation of mammalian cell growth by autocrine growth factors: analysis of consequences for inoculum cell density effects.Biotechnology and bioengineering, 33(11):1365–1378, 1989

    D Lauffenburger and C Cozens. Regulation of mammalian cell growth by autocrine growth factors: analysis of consequences for inoculum cell density effects.Biotechnology and bioengineering, 33(11):1365–1378, 1989

    1989

  4. [12]

    Bayesian decision-making shapes phenotypic landscapes from differentiation to cancer.Entropy, 28(3):312, 2026

    Arnab Barua and Haralampos Hatzikirou. Bayesian decision-making shapes phenotypic landscapes from differentiation to cancer.Entropy, 28(3):312, 2026

    2026

  5. [13]

    The free-energy principle: a unified brain theory?Nature reviews neuroscience, 11(2):127–138, 2010

    Karl Friston. The free-energy principle: a unified brain theory?Nature reviews neuroscience, 11(2):127–138, 2010

    2010

  6. [14]

    Collective behavior from surprise minimization

    C Heins, B Millidge, L Da Costa, R Mann, K Friston, and ID Couzin. Collective behavior from surprise minimization. arxiv.arXiv preprint arXiv:2307.14804, 2023

    2023

  7. [15]

    Gradientsensinginbayesianchemotaxis.Europhysicsletters,138(1):12001,2022

    AndreaAuconi,MajaNovak,andBenjaminMFriedrich. Gradientsensinginbayesianchemotaxis.Europhysicsletters,138(1):12001,2022

    2022

  8. [16]

    How a well-adapting immune system remembers

    Andreas Mayer, Vijay Balasubramanian, Aleksandra M Walczak, and Thierry Mora. How a well-adapting immune system remembers. Proceedings of the National Academy of Sciences, 116(18):8815–8823, 2019

    2019

  9. [17]

    Cambridge university press, 2023

    Simo Särkkä and Lennart Svensson.Bayesian filtering and smoothing, volume 17. Cambridge university press, 2023

    2023

  10. [18]

    Routledge, 2014

    Conrad Hal Waddington.The strategy of the genes. Routledge, 2014

    2014

  11. [19]

    Cellfatesashigh-dimensionalattractorstatesofacomplexgeneregulatory network.Physical review letters, 94(12):128701, 2005

    SuiHuang,GabrielEichler,YaneerBar-Yam,andDonaldEIngber. Cellfatesashigh-dimensionalattractorstatesofacomplexgeneregulatory network.Physical review letters, 94(12):128701, 2005

    2005

  12. [20]

    Geneticandnon-geneticinstabilityintumorprogression:linkbetweenthefitnesslandscapeandtheepigeneticlandscapeofcancer cells.Cancer and Metastasis Reviews, 32(3):423–448, 2013

    SuiHuang. Geneticandnon-geneticinstabilityintumorprogression:linkbetweenthefitnesslandscapeandtheepigeneticlandscapeofcancer cells.Cancer and Metastasis Reviews, 32(3):423–448, 2013

    2013

  13. [21]

    Ana M Soto, Giuseppe Longo, Maël Montévil, and Carlos Sonnenschein. The biological default state of cell proliferation with variation and motility, a fundamental principle for a theory of organisms.Progress in Biophysics and Molecular Biology, 122(1):16–23, 2016

    2016

  14. [22]

    KAGeiler-Samerotte,CRBauer,SLi,NZiv,DavidGresham,andMLSiegal.Thedetailsinthedistributions:whyandhowtostudyphenotypic variability.Current opinion in biotechnology, 24(4):752–759, 2013

    2013

  15. [23]

    Autocrine signaling can explain the emergence of allee effects in cancer cell populations.PLoS computational biology, 18(3):e1009844, 2022

    Philip Gerlee, Philipp M Altrock, Adam Malik, Cecilia Krona, and Sven Nelander. Autocrine signaling can explain the emergence of allee effects in cancer cell populations.PLoS computational biology, 18(3):e1009844, 2022

    2022

  16. [24]

    Qiuyang Zhang, Chunhua Yu, Sui Peng, Hao Xu, Ellen Wright, Xi Zhang, Xiaofang Huo, Edaire Cheng, Thai H Pham, Kiyotaka Asanuma, etal.Autocrinevegfsignalingpromotesproliferationofneoplasticbarrett’sepithelialcellsthroughaplc-dependentpathway.Gastroenterology, 146(2):461–472, 2014

    2014

  17. [25]

    Chapman and Hall/CRC, 2019

    Uri Alon.An introduction to systems biology: design principles of biological circuits. Chapman and Hall/CRC, 2019

    2019

  18. [26]

    The structure of bivariate distributions.The Annals of Mathematical Statistics, 29(3):719–736, 1958

    Henry Oliver Lancaster. The structure of bivariate distributions.The Annals of Mathematical Statistics, 29(3):719–736, 1958

    1958

  19. [27]

    Wound healing–aiming for perfect skin regeneration.Science, 276(5309):75–81, 1997

    Paul Martin. Wound healing–aiming for perfect skin regeneration.Science, 276(5309):75–81, 1997

    1997

  20. [28]

    Organ-size regulation in mammals.Cold Spring Harbor perspectives in biology, 7(9):a019240, 2015

    Alfredo I Penzo-Méndez and Ben Z Stanger. Organ-size regulation in mammals.Cold Spring Harbor perspectives in biology, 7(9):a019240, 2015

    2015

  21. [29]

    Size control: the developmental physiology of body and organ size regulation.Wiley Interdisciplinary Reviews: Developmental Biology, 4(4):335–356, 2015

    Rewatee H Gokhale and Alexander W Shingleton. Size control: the developmental physiology of body and organ size regulation.Wiley Interdisciplinary Reviews: Developmental Biology, 4(4):335–356, 2015

    2015

  22. [30]

    The hallmarks of cancer.cell, 100(1):57–70, 2000

    Douglas Hanahan and Robert A Weinberg. The hallmarks of cancer.cell, 100(1):57–70, 2000

    2000

  23. [31]

    Hallmarks of cancer: the next generation.cell, 144(5):646–674, 2011

    Douglas Hanahan and Robert A Weinberg. Hallmarks of cancer: the next generation.cell, 144(5):646–674, 2011. M.K Gupta et al. Page 17 of 34 Proliferation regulation

    2011

  24. [32]

    Thepi3k–aktnetworkattheinterfaceofoncogenicsignallingandcancermetabolism.NatureReviews Cancer, 20(2):74–88, 2020

    GertaHoxhajandBrendanDManning. Thepi3k–aktnetworkattheinterfaceofoncogenicsignallingandcancermetabolism.NatureReviews Cancer, 20(2):74–88, 2020

    2020

  25. [33]

    M.K Gupta et al

    GregoryJKimmel,SadeghMarzban,MehdiDamaghi,ArneTraulsen,AlexanderRAAnderson,JeffreyWest,andPhilippMAltrock.Universal principles of cell population growth follow from local contact inhibition.iScience, 29(6), 2026. M.K Gupta et al. Page 18 of 34 Proliferation regulation Suppleme...

    2026

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.