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arxiv: 2605.23722 · v1 · pith:YV5FA7TGnew · submitted 2026-05-22 · 🧮 math.DS

Sustained Limit Cycles in the Logistic Two-Gene Genetic Oscillator: A Delay-Driven Hopf Bifurcation

Ismail Belgacem This is my paper

Pith reviewed 2026-05-25 02:34 UTC · model grok-4.3

classification 🧮 math.DS
keywords delay differential equationsHopf bifurcationgenetic oscillatorlogistic repressionlimit cycletranscriptional delaynegative feedback loopp53-Mdm2
0
0 comments X

The pith

Delays in the logistic two-gene negative-feedback oscillator induce a Hopf bifurcation to sustained limit cycles once total delay exceeds an explicit critical value computed from the logistic derivatives.

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

The paper extends the standard logistic two-gene model by adding transcriptional delays and shows that the equilibrium, which is stable without delay, loses stability through a Hopf bifurcation as total delay crosses a closed-form threshold. The analysis supplies explicit formulas for the critical delay and frequency, proves that the loop-gain condition is necessary and sufficient, and establishes a uniform positive lower bound on the transversality condition. Global persistence of the bifurcation, supercriticality via Lindstedt-Poincaré reduction, and matching to p53-Mdm2 oscillation period within 3 percent are also obtained. The same framework yields closed-form results for cyclic N-gene loops.

Core claim

The equilibrium loses stability through a Hopf bifurcation as the total delay τ=τ1+τ2 crosses an explicit critical value τ_c; the Hopf frequency ω_c and τ_c are computed in closed form from the logistic derivatives; the loop-gain condition AB>γ1γ2 is necessary and sufficient; the transversality Re(dμ/dτ)|_τc>0 admits a parameter-uniform positive lower bound; and the bifurcation persists globally. A sum-of-delays symmetry reduces the analysis to the scalar parameter τ. For the symmetric-threshold loop supercriticality is proved by a Lindstedt-Poincaré reduction; for the general asymmetric loop the first Lyapunov coefficient and criticality criterion are obtained in closed form. The analysis,

What carries the argument

the characteristic equation of the delay-differential system whose roots cross the imaginary axis at the explicit Hopf pair (ω_c, τ_c) derived from the logistic repression derivatives

If this is right

  • The loop-gain condition AB > γ1 γ2 is necessary and sufficient for the delay-induced Hopf bifurcation to occur.
  • The Hopf period asymptote T ~ 2τ + C_∞ holds with explicit offset C_∞ independent of the specific logistic parameters.
  • The bifurcation is supercritical for the symmetric case and the first Lyapunov coefficient supplies an explicit criticality criterion for the asymmetric case.
  • The same closed-form transversality rate and delay-induced Hopf window extend directly to cyclic N-gene negative-feedback loops.

Where Pith is reading between the lines

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

  • Real genetic oscillators may rely on transcriptional delays to cross the stability threshold that the instantaneous model cannot reach.
  • The explicit τ_c formula supplies a direct test for whether measured delays in a given circuit are sufficient to produce oscillations.
  • The N-gene extension implies that longer loops can oscillate at smaller per-step delays provided the gain condition holds.

Load-bearing premise

The repression functions must be exactly logistic so that their equilibrium derivatives remain explicit and permit closed-form solution of the critical delay and frequency.

What would settle it

Measure the onset of sustained oscillations in a two-gene synthetic circuit whose transcriptional delays are tuned across the predicted τ_c while holding all other parameters fixed; the transition should occur at the computed value and the period should approach 2τ + C_∞.

read the original abstract

The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation clock, Hes1, p53) nevertheless rely on delays. We extend the logistic two-gene model to a delay-differential system with transcriptional delays $\tau_1$ and $\tau_2$, and prove that the equilibrium loses stability through a Hopf bifurcation as the total delay $\tau=\tau_1+\tau_2$ crosses an explicit critical value $\tau_c$. The Hopf frequency $\omega_c$ and $\tau_c$ are computed in closed form from the logistic derivatives; the loop-gain condition $AB>\gamma_1\gamma_2$ is necessary and sufficient; the transversality $\mathrm{Re}(d\mu/d\tau)|_{\tau_c}>0$ admits a parameter-uniform positive lower bound; and the bifurcation persists globally. A sum-of-delays symmetry reduces the analysis to the scalar parameter $\tau$. Numerical simulations confirm three regimes (damped, small limit cycle, relaxation), the supercritical amplitude scaling $A\sim c\sqrt{\tau-\tau_c}$, and the deep-relaxation period asymptote $T\sim 2\tau+C_\infty$ with closed-form offset $C_\infty$. For the symmetric-threshold loop, supercriticality is proved by a Lindstedt--Poincar\'e reduction yielding closed-form amplitude and frequency laws; for the general asymmetric loop it delivers a closed-form first Lyapunov coefficient and an explicit criticality criterion. Calibrated to p53--Mdm2 data, the closed-form Hopf period matches the observed oscillation within $3\%$, and the standard Hill-function model within a few percent. The analysis extends to cyclic $N$-gene loops, with a closed-form transversality rate valid for every $N$ and -- in the symmetric case -- an explicit delay-induced-Hopf window $\gamma^N<\Lambda<\gamma^N\sec^N(\pi/N)$.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes a two-gene negative-feedback genetic oscillator with logistic repression and transcriptional delays τ1, τ2. Via a sum-of-delays reduction it reduces the system to a single delay τ and proves that the unique equilibrium loses local asymptotic stability through a Hopf bifurcation when τ exceeds an explicit critical value τc. Closed-form expressions are given for the Hopf frequency ωc and τc from the characteristic equation (λ+γ1)(λ+γ2)+AB exp(−λτ)=0 under the necessary-and-sufficient loop-gain condition AB>γ1γ2; the transversality condition Re(dμ/dτ)|τc>0 is asserted to possess a parameter-uniform positive lower bound; supercriticality is proved via Lindstedt–Poincaré for the symmetric case and a closed-form first Lyapunov coefficient is supplied for the asymmetric case; global persistence of the limit cycle is claimed; numerical regimes, amplitude scaling, and period asymptotics are confirmed; and calibration to p53–Mdm2 data yields a Hopf period within 3% of observation. The analysis extends to cyclic N-gene loops with an explicit transversality rate for all N.

Significance. If the derivations hold, the paper supplies explicit, parameter-uniform analytical criteria for delay-induced Hopf bifurcations and supercritical limit cycles in a biologically motivated genetic circuit. The closed-form ωc, τc, first Lyapunov coefficient, and uniform transversality bound, together with the p53 calibration and the N-gene extension, constitute concrete strengths that could be directly usable for modeling circadian, segmentation-clock, or p53 oscillators.

major comments (2)
  1. [transversality analysis (implicit differentiation of characteristic equation)] The central claim that the equilibrium loses stability exactly when τ crosses the explicit τc for every biologically admissible parameter set rests on the assertion (abstract and transversality analysis) that Re(dμ/dτ)|τc admits a parameter-uniform positive lower bound. Implicit differentiation of the characteristic equation at λ=iωc produces an algebraically involved expression; the manuscript must exhibit either an explicit positive lower bound or a rigorous proof that the infimum over the region AB>γ1γ2, γi>0 is strictly positive. Without this, the uniform crossing statement is not secured.
  2. [global persistence claim] The global persistence statement for the limit cycle after the Hopf bifurcation is asserted without a cited theorem or section establishing the necessary a-priori bounds or Poincaré–Bendixson-type argument in the infinite-dimensional state space; this step is load-bearing for the claim of sustained oscillations.
minor comments (2)
  1. [p53 calibration paragraph] The 3% match to p53 data is stated in the abstract; the explicit parameter values used for the calibration and the precise definition of the observed period should be supplied in the main text or a table.
  2. [model section] Notation for the loop-gain quantities A, B and the decay rates γ1, γ2 should be introduced once with a single equation reference rather than re-defined in multiple sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying two points where the manuscript's claims require additional explicit support. Both comments are well-taken; we will strengthen the paper accordingly. Below we address each major comment and indicate the planned revisions.

read point-by-point responses

  1. Referee: [transversality analysis (implicit differentiation of characteristic equation)] The central claim that the equilibrium loses stability exactly when τ crosses the explicit τc for every biologically admissible parameter set rests on the assertion (abstract and transversality analysis) that Re(dμ/dτ)|τc admits a parameter-uniform positive lower bound. Implicit differentiation of the characteristic equation at λ=iωc produces an algebraically involved expression; the manuscript must exhibit either an explicit positive lower bound or a rigorous proof that the infimum over the region AB>γ1γ2, γi>0 is strictly positive. Without this, the uniform crossing statement is not secured.

    Authors: The referee correctly notes that the manuscript asserts a parameter-uniform positive lower bound for Re(dμ/dτ)|τc but does not display the full algebraic verification or the infimum argument. In the revision we will add an explicit lemma that (i) writes out the differentiated characteristic equation evaluated at λ = iωc, (ii) isolates the real part, and (iii) proves by direct estimation that this real part is bounded below by a strictly positive constant (independent of all admissible parameters) whenever AB > γ1γ2 and γi > 0. The bound will be stated in closed form and the proof will occupy a short dedicated subsection. revision: yes

  2. Referee: [global persistence claim] The global persistence statement for the limit cycle after the Hopf bifurcation is asserted without a cited theorem or section establishing the necessary a-priori bounds or Poincaré–Bendixson-type argument in the infinite-dimensional state space; this step is load-bearing for the claim of sustained oscillations.

    Authors: We agree that the global-persistence claim is stated without the supporting technical machinery. The revised manuscript will contain a new subsection that supplies the missing justification: we will derive uniform a-priori bounds on solutions via the method of steps and the logistic structure, then invoke a global Hopf-bifurcation theorem for retarded functional differential equations (with an appropriate reference) to conclude that the local supercritical cycle persists for all τ > τc. This will replace the current one-sentence assertion. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations from DDE characteristic equation via standard Hopf analysis are self-contained.

full rationale

The paper starts from the delay-differential equations, reduces via sum-of-delays symmetry to a scalar τ, and applies classical characteristic-equation analysis to obtain the explicit τ_c, ω_c, and loop-gain condition AB > γ1γ2 as necessary and sufficient. The transversality Re(dμ/dτ)|_τc > 0 and its claimed parameter-uniform lower bound are obtained by implicit differentiation of the characteristic equation at λ = iω_c; this is an algebraic derivation, not a fit or self-definition. The first Lyapunov coefficient for the asymmetric case and the Lindstedt-Poincaré reduction for the symmetric case are likewise computed directly from the model. The p53 calibration is presented as an external consistency check, not part of the bifurcation proof. No self-citation is load-bearing, no ansatz is smuggled, and no quantity is renamed as a prediction. The derivation chain is independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The analysis rests on the logistic form of the repression functions (allowing explicit derivatives), the assumption of constant transcriptional delays, and standard theorems of delay-differential equations; no new entities are postulated and no parameters are fitted inside the bifurcation proof itself.

axioms (3)
  • domain assumption The non-delayed Jacobian has uniformly negative trace for all biological parameter values.
    Invoked as the starting point that makes the delay-free equilibrium stable.
  • domain assumption Transcriptional delays τ1 and τ2 are constant and enter only as pure time shifts in the argument of the repression functions.
    The model extension is defined with these constant delays; the sum-of-delays symmetry is used to reduce to a single parameter τ.
  • domain assumption The repression functions admit closed-form derivatives at equilibrium that permit an explicit characteristic equation.
    Required for the closed-form expressions of ω_c and τ_c from the logistic derivatives.

pith-pipeline@v0.9.0 · 5902 in / 1848 out tokens · 27446 ms · 2026-05-25T02:34:59.887348+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 2 internal anchors

  1. [1]

    Exploring Logistic Functions as Robust Alternatives to Hill Functions in Genetic Network Modeling

    I. Belgacem,Exploring Logistic Functions as Robust Alternatives to Hill Functions in Genetic Network Modeling, arXiv:2512.14325 [math.DS], 2025. Available at: https://arxiv.org/abs/2512.14325

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  2. [2]

    Logistic Gene Regulatory Networks: Prevention of Expression Shutdown, and Numerical Stability Beyond Hill Function

    I. Belgacem,Logistic Gene Regulatory Networks: Prevention of Expression Shutdown, and Numerical Stability Beyond Hill Functions, arXiv preprint arXiv:2605.01056, 2026. Available at: https://arxiv.org/abs/2605.01056

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    P. E. Hardin, J. C. Hall, M. Rosbash, Feedback of theDrosophilaperiod gene product on circadian cycling of its messenger RNA levels,Nature343(1990) 536–540

    work page 1990

  4. [4]

    J. C. Dunlap, J. J. Loros, P. J. DeCoursey,Chronobiology: Biological Timekeeping, Sinauer Associates, 2004. 48

    work page 2004

  5. [5]

    Lewis, Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator,Current Biology13(2003) 1398–1408

    J. Lewis, Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator,Current Biology13(2003) 1398–1408

    work page 2003

  6. [6]

    Hirata, S

    H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa, R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop,Science298(2002) 840–843

    work page 2002

  7. [7]

    N. A. M. Monk, Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays,Current Biology13(2003) 1409–1413

    work page 2003

  8. [8]

    Hoffmann, A

    A. Hoffmann, A. Levchenko, M. L. Scott, D. Baltimore, The IκB–NF-κB signaling module: temporal control and selective gene activation,Science298(2002) 1241– 1245

    work page 2002

  9. [9]

    D. E. Nelsonet al., Oscillations in NF-κB signalling control the dynamics of gene expression,Science306(2004) 704–708

    work page 2004

  10. [10]

    Geva-Zatorsky, N

    N. Geva-Zatorsky, N. Rosenfeld, S. Itzkovitz, R. Milo, A. Sigal, E. Dekel, T. Yarnitzky, Y. Liron, P. Polak, G. Lahav, U. Alon, Oscillations and vari- ability in the p53 system,Molecular Systems Biology2(2006) 2006.0033, DOI: 10.1038/msb4100068

    work page doi:10.1038/msb4100068 2006

  11. [11]

    M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science197(1977) 287–289

    work page 1977

  12. [12]

    Glass, M

    L. Glass, M. C. Mackey,From Clocks to Chaos: The Rhythms of Life, Princeton University Press, 1988

    work page 1988

  13. [13]

    K. L. Cooke, P. van den Driessche, On zeroes of some transcendental equations, Funkcialaj Ekvacioj29(1986) 77–90

    work page 1986

  14. [14]

    Kuang,Delay Differential Equations With Applications in Population Dynam- ics, Mathematics in Science and Engineering, Vol

    Y. Kuang,Delay Differential Equations With Applications in Population Dynam- ics, Mathematics in Science and Engineering, Vol. 191, Academic Press, 1993

    work page 1993

  15. [15]

    H. L. Smith,An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Vol. 57, Springer, 2011

    work page 2011

  16. [16]

    J. K. Hale, S. M. Verduyn Lunel,Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer, 1993

    work page 1993

  17. [17]

    Beretta, Y

    E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential sys- tems with delay-dependent parameters,SIAM Journal on Mathematical Analysis 33(2002) 1144–1165

    work page 2002

  18. [18]

    B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan,Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, Vol. 41, Cambridge University Press, 1981. 49

    work page 1981

  19. [19]

    Young, H

    R. Young, H. Bremer, Polypeptide-chain-elongation rate inEscherichia coliB/r as a function of growth rate,Biochemical Journal160(1976) 185–194

    work page 1976

  20. [20]

    Ansmann, Efficiently and easily integrating differential equations with JiT- CODE, JiTCDDE, and JiTCSDE,Chaos: An Interdisciplinary Journal of Nonlinear Science28(2018) 043116

    G. Ansmann, Efficiently and easily integrating differential equations with JiT- CODE, JiTCDDE, and JiTCSDE,Chaos: An Interdisciplinary Journal of Nonlinear Science28(2018) 043116

    work page 2018

  21. [21]

    Soetaert, T

    K. Soetaert, T. Petzoldt, R. W. Setzer, Solving differential equations in R: packagedeSolve,Journal of Statistical Software33(9) (2010) 1–25, DOI: 10.18637/jss.v033.i09

    work page doi:10.18637/jss.v033.i09 2010

  22. [22]

    MacDonald,Biological Delay Systems: Linear Stability Theory, Cambridge University Press, 1989

    N. MacDonald,Biological Delay Systems: Linear Stability Theory, Cambridge University Press, 1989

    work page 1989

  23. [23]

    S. P. Hastings, J. J. Tyson, D. Webster, Existence of periodic solutions for nega- tive feedback cellular control systems,Journal of Differential Equations25(1977) 39–64

    work page 1977

  24. [24]

    J. M. Mahaffy, C. V. Pao, Models of genetic control by repression with time delays and spatial effects,Journal of Mathematical Biology20(1984) 39–57

    work page 1984

  25. [25]

    Smolen, D

    P. Smolen, D. A. Baxter, J. H. Byrne, A reduced model clarifies the role of feedback loops and time delays in theDrosophilacircadian oscillator,Biophysical Journal83(2002) 2349–2359

    work page 2002

  26. [26]

    M. Xiao, J. Cao, Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays,Mathematical Biosciences215(2008) 55–63

    work page 2008

  27. [27]

    Verdugo, R

    A. Verdugo, R. Rand, Hopf bifurcation in a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation13(2008) 235–242

    work page 2008

  28. [28]

    R. Rand, A. Verdugo, Hopf bifurcation formula for first order differential-delay equations,Communications in Nonlinear Science and Numerical Simulation12 (2007) 859–864

    work page 2007

  29. [29]

    G. Wang, Z. Yang, M. Turcotte, Dynamic analysis of the time-delayed genetic regulatory network between two auto-regulated and mutually inhibitory genes, Bulletin of Mathematical Biology82(2020) 46

    work page 2020

  30. [30]

    Wu, Symmetric functional differential equations and neural networks with memory,Transactions of the American Mathematical Society350(1998) 4799– 4838

    J. Wu, Symmetric functional differential equations and neural networks with memory,Transactions of the American Mathematical Society350(1998) 4799– 4838

    work page 1998

  31. [31]

    Huang, J

    C. Huang, J. Cao, Hopf bifurcation in ann-dimensional Goodwin model via multiple delays feedback,Nonlinear Dynamics79(2015) 2541–2552, DOI: 10.1007/s11071-014-1831-1. 50

    work page doi:10.1007/s11071-014-1831-1 2015

  32. [32]

    Belgacem, R

    I. Belgacem, R. Edwards, E. Farcot, Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit,Chaos: An Interdisciplinary Journal of Nonlinear Science35(5) (2025) 053151, DOI: 10.1063/5.0243955

    work page doi:10.1063/5.0243955 2025

  33. [33]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, R. Edwards, Control of negative feedback loops in genetic networks, in:59th IEEE Conference on Decision and Control (CDC), Jeju Island, Republic of Korea, IEEE, 2020, DOI: 10.1109/CDC42340.2020.9304088

    work page doi:10.1109/cdc42340.2020.9304088 2020

  34. [34]

    Chambon, I

    L. Chambon, I. Belgacem, J.-L. Gouz´ e, Qualitative control of undesired oscil- lations in a genetic negative feedback loop with uncertain measurements, Automatica112(2020) 108642

    work page 2020

  35. [35]

    Farcot, S

    E. Farcot, S. Best, R. Edwards, I. Belgacem, X. Xu, P. Gill, Chaos in a ring circuit, Chaos: An Interdisciplinary Journal of Nonlinear Science29(2019) 043103

    work page 2019

  36. [36]

    Belgacem, H

    I. Belgacem, H. Bensalah, B. Cherki, R. Edwards, The probabilistic con- volution regularization of Zeno hybrid systems, in:18th European Con- trol Conference (ECC), Naples, Italy, IEEE, 2019, pp. 3527–3532, DOI: 10.23919/ECC.2019.8795626

    work page doi:10.23919/ecc.2019.8795626 2019

  37. [37]

    Belgacem, S

    I. Belgacem, S. Casagranda, E. Grac, D. Ropers, J.-L. Gouz´ e, Reduction and sta- bility analysis of a transcription–translation model of RNA polymerase,Bulletin of Mathematical Biology80(2018) 294–318

    work page 2018

  38. [38]

    Belgacem, E

    I. Belgacem, E. Grac, D. Ropers, J.-L. Gouz´ e, Stability analysis of a reduced transcription–translation model of RNA polymerase, in:53rd IEEE Conference on Decision and Control (CDC), Los Angeles, USA, IEEE, 2014, pp. 3924–3929, DOI: 10.1109/CDC.2014.7039999

    work page doi:10.1109/cdc.2014.7039999 2014

  39. [39]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, Mathematical study of the global dynamics of a concave gene expression model, in:22nd Mediterranean Conference on Control and Automation (MED’14), Palermo, Italy, IEEE, 2014, pp. 1341–1346, DOI: 10.1109/MED.2014.6961562

    work page doi:10.1109/med.2014.6961562 2014

  40. [40]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, Stability analysis and reduction of gene transcription models, in:52nd IEEE Conference on Decision and Control (CDC), Florence, Italy, IEEE, 2013, pp. 2691–2696, DOI: 10.1109/CDC.2013.6760289

    work page doi:10.1109/cdc.2013.6760289 2013

  41. [41]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, Global stability of enzymatic chains of full reversible Michaelis–Menten reactions,Acta Biotheoretica61(2013) 425–436

    work page 2013

  42. [42]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, Analysis and reduction of transcription translation coupled models for gene expression, in:12th IFAC Symposium on Computer Applications in Biotechnology (CAB), Mumbai, India, Elsevier, 2013, pp. 42–47, DOI: 10.3182/20131216-3-IN-2044.00012. 51

    work page doi:10.3182/20131216-3-in-2044.00012 2013

  43. [43]

    Belgacem, J.-L

    I. Belgacem, J.-L. Gouz´ e, Global stability of full open reversible Michaelis–Menten reactions, in:8th IFAC Symposium on Advanced Control of Chemical Processes, Singapore, Elsevier, 2012, pp. 591–596, DOI: 10.3182/20120710-4-SG-2026.00039

    work page doi:10.3182/20120710-4-sg-2026.00039 2012

  44. [44]

    Lev Bar-Or, R

    R. Lev Bar-Or, R. Maya, L. A. Segel, U. Alon, A. J. Levine, M. Oren, Generation of oscillations by the p53–Mdm2 feedback loop: A theoretical and experimen- tal study,Proceedings of the National Academy of Sciences USA97(21) (2000) 11250–11255, DOI: 10.1073/pnas.210171597. 52

    work page doi:10.1073/pnas.210171597 2000