arxiv: 2605.01056 · v1 · submitted 2026-05-01 · 🧬 q-bio.MN · math.DS

Recognition: unknown

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

Ismail Belgacem

Authors on Pith no claims yet

Pith reviewed 2026-05-09 14:34 UTC · model grok-4.3

classification 🧬 q-bio.MN math.DS

keywords gene regulatory networkslogistic functionsHill functionsnumerical stabilitybistabilitynegative feedback oscillatorBoolean networksgene expression

0 comments

The pith

Logistic functions fix the smoothness, numerical stability, and zero-basal-rate flaws of Hill functions in gene regulatory network models.

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

The paper establishes that logistic activation functions provide a globally smooth, real-valued, and strictly positive-at-zero replacement for Hill functions when modeling gene regulation. These properties eliminate three problems that arise with non-integer cooperativity exponents: loss of global differentiability, silent insertion of complex numbers into ODE trajectories, and complete shutdown of basal production that traps bistable circuits in inactive states. A reader would care because the change enables provable stability in oscillators, finite escape times from off-states in positive autoregulation, and reliable long-term integration of Boolean-derived networks without solver warnings or failures. Demonstrations cover local stability via Routh-Hurwitz, explicit saddle-node thresholds, automatic De Morgan translation of an 11-gene cell-cycle network, and stable runs to t=200 on an 80-gene system.

Core claim

Logistic functions f±, defined to be globally C^∞, real for every real input, and strictly positive at zero, resolve the three structural flaws of Hill functions simultaneously. In a two-gene negative-feedback oscillator, the Routh-Hurwitz criterion shows local asymptotic stability for all positive parameter values and rules out Hopf bifurcation without time delays. For bistable positive autoregulation, saddle-node thresholds are given by explicit transcendental equations; with E. coli parameters the logistic basal rate produces off-state escape in approximately 44 minutes while the Hill model remains trapped indefinitely. The product-of-logistics De Morgan formalism translates Boolean logic

What carries the argument

The logistic activation functions f± that replace Hill functions, guaranteeing global smoothness, real arithmetic, and positive basal production rate.

Load-bearing premise

That logistic functions with positive basal production represent the underlying biology at least as accurately as Hill functions for the gene circuits and parameter ranges considered.

What would settle it

A laboratory measurement showing that a bistable positive-autoregulation circuit remains permanently off under conditions where the logistic model predicts escape within one hour, or a simulation run in which a Hill model with non-integer exponent produces no complex values or warnings.

Figures

Figures reproduced from arXiv: 2605.01056 by Ismail Belgacem.

**Figure 1.** Figure 1: Architecture of a two-gene negative feedback loop. Gene A activates gene B (blue arrow), while gene B represses gene A (red bar) view at source ↗

**Figure 2.** Figure 2: Temporal evolution of the two-gene oscillator system (4). Parameters: 𝜆 = 3, 𝜅1 = 3, 𝛾1 = 0.25, 𝜅2 = 4, 𝛾2 = 0.5, 𝜃1 = 4, 𝜃2 = 3; initial conditions 𝑥01 = 𝑥02 = 1 view at source ↗

**Figure 3.** Figure 3: Temporal evolution of the 11-gene Traynard cell-cycle logistic ODE system over 𝑡 ∈ [0, 60], starting from the initial conditions in view at source ↗

**Figure 4.** Figure 4: Comparison of the decreasing logistic functions 𝑓 − 1 (𝑥, 𝜃, 𝜆) = 1∕(1 + 𝑒 −𝜆(𝑤𝑥+𝜃) ) (our model) and 𝑓 − 2 (𝑥, 𝜃, 𝜇) = 1∕(1+exp(−𝜇(𝑤𝑥−𝜃))) (Samuilik) with 𝑤 = −1, alongside the decreasing Hill function ℎ − (𝑥, 𝜃, 𝑛) = 𝜃 𝑛∕(𝑥 𝑛+𝜃 𝑛 ). Parameters: 𝑛 = 4, 𝜃 = 3, 𝜆 = 𝑛∕𝜃 ≈ 1.333, 𝜇 = 𝜆. The critical point of 𝑓 − 1 is 𝑥𝑐 = −𝜃∕𝑤 = 3 > 0 (biologically meaningful), while that of 𝑓 − 2 is 𝑥𝑐 = 𝜃∕𝑤 = −3 < 0 (outsid… view at source ↗

**Figure 5.** Figure 5: Trajectories of the genetic oscillator under small perturbations and strong degradation. The logistic model (solid) escapes the low-expression trap, while the Hill model (dashed) stagnates. Left: 𝑥1 (𝑡); Right: 𝑥2 (𝑡). First Author et al.: Page 39 of 37 view at source ↗

**Figure 6.** Figure 6: Protein dynamics in positive autoregulation under low initial conditions (𝑚(0) = 0.01, 𝑥(0) = 0.01) with feedback amplification 𝛼 ≈ 600. Logistic model (monostable-high regime): The system escapes the off-state in approximately 2650 s (∼44 min) due to basal production 𝑘𝑚𝑓(0) ≈ 0.000142 s −1, reaching 𝑥 ≈ 38 at 10,000 s and approaching 𝑥ss ≈ 600 asymptotically. Hill model (bistable regime): mRNA decays expo… view at source ↗

**Figure 7.** Figure 7: Hill-function ODE system (𝑛 ≈ 3.51, non-integer), full time horizon 𝑡 ∈ [0, 200]. Trajectories of all 80 state variables. The solver’s reliable domain ends between 𝑡 = 50 and 𝑡 = 100: beyond that point several curves diverge catastrophically. The 𝑦-axis spans [−500 000, 2 × 106 ]; the most divergent variable (blue curve, 𝑥2 ) reaches ≈ 2 × 106 at 𝑡 = 200, while the most negative variable descends to approx… view at source ↗

**Figure 8.** Figure 8: Hill-function ODE system (𝑛 ≈ 3.51, non-integer), early window 𝑡 ∈ [0, 65]. The same simulation as view at source ↗

**Figure 9.** Figure 9: Logistic-function ODE system (same 𝑛 and 𝜃 parameters), full time horizon 𝑡 ∈ [0, 200]. Trajectories of all 80 state variables. NDSolve completes the integration without any warnings. All variables remain strictly non-negative throughout and converge to bounded steady states; the 𝑦-axis is confined to [0, 275], consistent with the biological bound 𝜅𝑖∕𝛾𝑖 . Most variables settle before 𝑡 = 50. Two variables … view at source ↗

read the original abstract

Hill functions, the standard tool for modelling gene regulatory networks, carry three structural flaws when the cooperativity exponent is non-integer: loss of global smoothness, silent complex-valued arithmetic corruption of ODE trajectories, and an identically zero basal production rate that traps bistable models in off-states. Logistic functions $f^\pm$, being globally $C^\infty$, real-valued for all arguments, and strictly positive at zero, resolve all three simultaneously. For a two-gene negative-feedback oscillator, local asymptotic stability is established for all positive parameters via the Routh--Hurwitz criterion, and no Hopf bifurcation is possible without time delays. For bistable positive autoregulation, saddle-node thresholds are characterised through explicit transcendental equations; with biophysically grounded \textit{E.~coli} parameters, basal logistic production drives off-state escape in $\approx 44$~min while the Hill model remains permanently trapped. The 11-gene Traynard cell-cycle Boolean network is translated automatically via the product-of-logistics De~Morgan formalism and integrated without warnings, all variables remaining bounded and non-negative. The De~Morgan framework places every repressor threshold at a positive measurable concentration, whereas the weighted-sum formulation of Samuilik et al.\ places repressor critical points at negative concentrations, rendering them biologically inert. On an 80-gene Boolean-derived ODE system with $n = 3.509$, the Hill solver entered silent complex-valued contamination at $t \approx 52.64$ and terminated near $t \approx 63$--$65$; the logistic formulation completed $t \in [0, 200]$ without a single warning. The always-positive production rate ensures full controllability, enabling sliding mode, model predictive, and feedback-linearisation strategies where Hill-based formulations fail.

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

Logistic functions fix numerical headaches in GRN ODEs but trade one modeling choice for another on basal expression.

read the letter

The paper's core move is replacing Hill functions with logistic ones in gene regulatory network ODEs. This keeps everything smooth and real-valued while adding a small positive basal production rate at zero input. The author shows this sidesteps three documented problems: non-integer cooperativity breaking global differentiability, silent complex-value creep in solvers, and permanent trapping in off-states for bistable positive feedback loops. Concrete checks include Routh-Hurwitz stability for a two-gene oscillator (no Hopf without delay), explicit saddle-node conditions for positive autoregulation, and an 80-gene Boolean-derived system that runs cleanly to t=200 under logistic but hits complex contamination under Hill around t=52. The De Morgan product form for repressor logic also keeps thresholds at positive concentrations, unlike some weighted-sum alternatives. These are useful, reproducible numerical results for anyone who has watched a large GRN simulation die on arithmetic warnings. The E. coli bistable example reports off-state escape in roughly 44 minutes under grounded parameters, which follows directly from the logistic property f(0)>0. That is the part worth pausing on. The escape is not an extra feature; it is the direct consequence of replacing a zero basal rate with a positive one. If the real promoters in question have basal transcription consistent with zero within experimental error, the logistic version is changing the model rather than repairing it. The paper grounds parameters biophysically but does not show side-by-side comparison against measured escape times or basal rates, so the claim that this constitutes prevention rather than distortion rests on that modeling choice. The 11-gene cell-cycle translation and the large-network run are still solid evidence that the formulation integrates without warnings. This work is aimed at computational systems biologists who simulate large or stiff GRNs and have hit the Hill-function limitations in practice. The numerical demonstrations and small-system proofs are clear enough that a serious editor should send it to referees, mainly to check the parameter grounding and to ask whether the logistic leak needs explicit validation against data before it is adopted as a default replacement.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes logistic functions f^± as replacements for Hill functions in gene regulatory network (GRN) models. It argues that logistics simultaneously resolve three Hill-function flaws for non-integer cooperativity: loss of global C^∞ smoothness, silent complex-valued arithmetic corruption of ODE trajectories, and identically zero basal production that traps bistable models in off-states. The paper establishes local asymptotic stability of a two-gene negative-feedback oscillator for all positive parameters via the Routh-Hurwitz criterion with no Hopf bifurcation possible without delays; derives saddle-node thresholds for bistable positive autoregulation via explicit transcendental equations; reports off-state escape in ≈44 min under biophysically grounded E. coli parameters; translates an 11-gene Traynard cell-cycle Boolean network via product-of-logistics De Morgan formalism; and demonstrates that an 80-gene Boolean-derived ODE system (n=3.509) completes t∈[0,200] without warnings while the Hill formulation enters complex-valued contamination at t≈52.64.

Significance. If the logistic basal rate is biologically appropriate, the work supplies a globally smooth, real-valued, and fully controllable framework for GRN modeling that avoids numerical artifacts and enables control-theoretic methods. The Routh-Hurwitz proof, explicit transcendental saddle-node conditions, and reproducible numerical demonstrations on both small and 80-gene systems constitute clear strengths. The De Morgan threshold placement at positive concentrations is a useful contrast to prior weighted-sum formulations. Significance is tempered by the need for stronger justification that non-zero basal production improves rather than distorts bistable dynamics.

major comments (2)

[Bistable positive autoregulation] Bistable positive autoregulation section: the claim that logistic f^± 'prevents expression shutdown' rests on the finite mean escape time produced exactly by f(0)>0. The manuscript reports ≈44 min escape under 'biophysically grounded E. coli parameters' but does not list the explicit basal production rate, its literature source, or sensitivity analysis; without these, it is impossible to assess whether the result reflects measured promoter leakiness or post-hoc parameter selection that artifactually destabilizes the off-state.
[Two-gene negative-feedback oscillator] Oscillator stability section: the Routh-Hurwitz criterion is applied to conclude local asymptotic stability for all positive parameters and no Hopf bifurcation without delays. The characteristic equation and the explicit Routh array entries should be provided so readers can verify the sign conditions independently; their absence makes the 'for all positive parameters' claim difficult to check.

minor comments (2)

[Introduction / Methods] The explicit functional forms of the logistic activation f^+ and repression f^- (including the precise definition of the cooperativity parameter) are referenced but not displayed in the abstract or early sections; they should appear with equation numbers before any stability or bifurcation analysis.
[Large-network numerical integration] In the 80-gene numerical example the cooperativity n=3.509 is stated without indicating whether it arises from a fit to data or is chosen for illustration; this detail affects reproducibility of the complex-value failure time t≈52.64.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications and supporting material.

read point-by-point responses

Referee: [Bistable positive autoregulation] Bistable positive autoregulation section: the claim that logistic f^± 'prevents expression shutdown' rests on the finite mean escape time produced exactly by f(0)>0. The manuscript reports ≈44 min escape under 'biophysically grounded E. coli parameters' but does not list the explicit basal production rate, its literature source, or sensitivity analysis; without these, it is impossible to assess whether the result reflects measured promoter leakiness or post-hoc parameter selection that artifactually destabilizes the off-state.

Authors: We agree that explicit documentation is required. In the revised manuscript we will state the precise basal production rate employed for the E. coli parameter set, cite the primary literature sources for measured promoter leakiness in E. coli, and add a sensitivity analysis that varies the basal rate over the range of experimentally reported values. This will show that off-state escape times remain on the order of tens of minutes for biophysically plausible leakiness levels. revision: yes
Referee: [Two-gene negative-feedback oscillator] Oscillator stability section: the Routh-Hurwitz criterion is applied to conclude local asymptotic stability for all positive parameters and no Hopf bifurcation without delays. The characteristic equation and the explicit Routh array entries should be provided so readers can verify the sign conditions independently; their absence makes the 'for all positive parameters' claim difficult to check.

Authors: We will include the full derivation in the revision. The updated section will present the Jacobian of the two-gene system, the resulting characteristic polynomial, and the complete Routh array with all entries and sign conditions. This will allow direct verification that the Hurwitz criteria are satisfied for every positive parameter combination. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper derives local stability for the negative-feedback oscillator directly from the Routh-Hurwitz criterion applied to the logistic ODE system, obtains saddle-node thresholds for positive autoregulation via explicit transcendental equations, and performs direct numerical integration on the 11-gene and 80-gene networks. All results follow from the stated functional forms of f^±, the De Morgan translation rule, and externally supplied biophysical parameters without any fitting of outputs to inputs, self-referential definitions, or load-bearing self-citations. The contrast with Hill functions is presented as a transparent consequence of the basal-rate difference rather than a derived prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of the logistic sigmoid and its application to GRN ODEs; no new entities are postulated.

axioms (1)

standard math The logistic function is globally C^∞, real-valued for all real arguments, and strictly positive at zero
Invoked throughout the abstract as the basis for resolving the three Hill-function flaws.

pith-pipeline@v0.9.0 · 5622 in / 1343 out tokens · 45414 ms · 2026-05-09T14:34:04.551615+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

82 extracted references · 13 canonical work pages · 1 internal anchor

[1]

An empirical extremum principle for the hill coefficient in ligand-protein interactions showing negative cooperativity

Abeliovich, H., 2005. An empirical extremum principle for the hill coefficient in ligand-protein interactions showing negative cooperativity. Biophysical journal 89, 76–79

2005
[2]

Atheoreticalexplorationofbirhythmicityinthep53-mdm2network

Abou-Jaoudé,W.,Chaves,M.,Gouzé,J.L.,2011. Atheoreticalexplorationofbirhythmicityinthep53-mdm2network. PLOSone6,e17075

2011
[3]

Enhancementofcellularmemorybyreducingstochastictransitions

Acar,M.,Becskei,A.,VanOudenaarden,A.,2005. Enhancementofcellularmemorybyreducingstochastictransitions. Nature435,228–232

2005
[4]

Quantitativemodelforgeneregulationbylambdaphagerepressor

Ackers,G.K.,Johnson,A.D.,Shea,M.A.,1982. Quantitativemodelforgeneregulationbylambdaphagerepressor. Proceedingsofthenational academy of sciences 79, 1129–1133

1982
[5]

The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in drosophila melanogaster

Albert, R., Othmer, H.G., 2003. The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in drosophila melanogaster. Journal of theoretical biology 223, 1–18

2003
[6]

An introduction to systems biology

Alon, U., 2007. An introduction to systems biology. Chapman & Hall/CRC, Boca Raton

2007
[7]

Cellular heterogeneity: do differences make a difference? Cell 141, 559–563

Altschuler, S.J., Wu, L.F., 2010. Cellular heterogeneity: do differences make a difference? Cell 141, 559–563

2010
[8]

Engineering stability in gene networks by autoregulation

Becskei, A., Serrano, L., 2000. Engineering stability in gene networks by autoregulation. Nature 405, 590–593

2000
[9]

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

Belgacem, I., 2025. Exploring logistic functions as robust alternatives to hill functions in genetic network modeling. arXiv preprint arXiv:2512.14325

work page internal anchor Pith review Pith/arXiv arXiv 2025
[10]

Theprobabilisticconvolutionregularizationofzenohybridsystems,in:201918th European Control Conference (ECC), IEEE

Belgacem,I.,Bensalah,H.,Cherki,B.,Edwards,R.,2019. Theprobabilisticconvolutionregularizationofzenohybridsystems,in:201918th European Control Conference (ECC), IEEE. pp. 750–757

2019
[11]

Reduction and stability analysis of a transcription–translation model of rna polymerase

Belgacem, I., Casagranda, S., Grac, E., Ropers, D., Gouzé, J.L., 2018. Reduction and stability analysis of a transcription–translation model of rna polymerase. Bulletin of Mathematical Biology 80, 294–318

2018
[12]

Computer-aidedanalysisofhigh-dimensionalGlassnetworks:periodicity,chaos,andbifurcations in a ring circuit

Belgacem,I.,Edwards,R.,Farcot,E.,2025. Computer-aidedanalysisofhigh-dimensionalGlassnetworks:periodicity,chaos,andbifurcations in a ring circuit. Chaos: An Interdisciplinary Journal of Nonlinear Science doi:10.1063/5.0243955

work page doi:10.1063/5.0243955 2025
[13]

Global stability of full open reversible Michaelis–Menten reactions, in: 8th IFAC Symposium on Advanced Control of Chemical Processes, Singapore

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

work page doi:10.3182/20120710-4-sg-2026.00039 2012
[14]

Analysis and reduction of transcription translation coupled models for gene expression

Belgacem, I., Gouzé, J.L., 2013a. Analysis and reduction of transcription translation coupled models for gene expression. IFAC Proceedings Volumes 46, 36–41
[15]

Global stability of enzymatic chains of full reversible michaelis-menten reactions

Belgacem, I., Gouzé, J.L., 2013b. Global stability of enzymatic chains of full reversible michaelis-menten reactions. Acta biotheoretica 61, 425–436
[16]

Stabilityanalysisandreductionofgenetranscriptionmodels,in:IEEE52ndAnnualConferenceonDecision and Control (CDC’13), Florence, Italy

Belgacem,I.,Gouzé,J.L.,2013. Stabilityanalysisandreductionofgenetranscriptionmodels,in:IEEE52ndAnnualConferenceonDecision and Control (CDC’13), Florence, Italy. pp. 2691–2696. doi:10.1109/CDC.2013.6760289

work page doi:10.1109/cdc.2013.6760289 2013
[17]

Mathematical study of the global dynamics of a concave gene expression model, in: 22nd Mediterranean Conference on Control and Automation (MED’14), Palermo, Italy

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

work page doi:10.1109/med.2014.6961562 2014
[18]

Control of negative feedback loops in genetic networks, in: 2020 59th IEEE Conference on Decision and Control (CDC), IEEE

Belgacem, I., Gouzé, J.L., Edwards, R., 2020. Control of negative feedback loops in genetic networks, in: 2020 59th IEEE Conference on Decision and Control (CDC), IEEE. pp. 5098–5105

2020
[19]

Overlapping

Belgacem, I., Gouzé, J.L., Edwards, R., 2021. Control of negative feedback loops in genetic networks, in: Proceedings of the 59th IEEE Conference on Decision and Control (CDC), Jeju Island, Republic of Korea. doi:10.1109/CDC42340.2020.9304088

work page doi:10.1109/cdc42340.2020.9304088 2021
[20]

Stability analysis of a reduced transcription-translation model of RNA polymerase, in: IEEE 53rd Conference on Decision and Control (CDC), Los Angeles, California, USA

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

work page doi:10.1109/cdc.2014.7039999 2014
[21]

Quantification of protein half-lives in the budding yeast proteome

Belle, A., Tanay, A., Bitincka, L., Shamir, R., O’Shea, E.K., 2006. Quantification of protein half-lives in the budding yeast proteome. Proceedings of the National Academy of Sciences 103, 13004–13009

2006
[22]

Modeling and analysis of gene regulatory networks, in: Modeling in computational biology and biomedicine: A multidisciplinary endeavor

Bernot, G., Comet, J.P., Richard, A., Chaves, M., Gouzé, J.L., Dayan, F., 2012. Modeling and analysis of gene regulatory networks, in: Modeling in computational biology and biomedicine: A multidisciplinary endeavor. Springer, pp. 47–80

2012
[23]

Global analysis of mrna decay and abundance in escherichia coliatsingle-generesolutionusingtwo-colorfluorescentdnamicroarrays

Bernstein, J.A., Khodursky, A.B., Lin, P.H., Lin-Chao, S., Cohen, S.N., 2002. Global analysis of mrna decay and abundance in escherichia coliatsingle-generesolutionusingtwo-colorfluorescentdnamicroarrays. ProceedingsoftheNationalAcademyofSciences99,9697–9702. First Author et al.: Page 35 of 37 Short Title of the Article

2002
[24]

Causal reasoning on boolean control networks based on abduction: theory and application to cancer drug discovery

Biane, C., Delaplace, F., 2018. Causal reasoning on boolean control networks based on abduction: theory and application to cancer drug discovery. IEEE/ACM transactions on computational biology and bioinformatics 16, 1574–1585

2018
[25]

Transcriptionalregulationbythenumbers:models

Bintu,L.,Buchler,N.E.,Garcia,H.G.,Gerland,U.,Hwa,T.,Kondev,J.,Phillips,R.,2005. Transcriptionalregulationbythenumbers:models. Current opinion in genetics & development 15, 116–124

2005
[26]

Hill function-based models of transcriptional switches: impact of specific, nonspecific, functional and nonfunctional binding

Bottani, S., Veitia, R.A., 2017. Hill function-based models of transcriptional switches: impact of specific, nonspecific, functional and nonfunctional binding. Biological Reviews 92, 953–963

2017
[27]

Qualitativecontrolofundesiredoscillationsinageneticnegativefeedbackloopwithuncertain measurements

Chambon,L.,Belgacem,I.,Gouzé,J.L.,2020. Qualitativecontrolofundesiredoscillationsinageneticnegativefeedbackloopwithuncertain measurements. Automatica 112, 108642

2020
[28]

The properties of logistic function and applications to neural network approximation

Chen, Z., Cao, F., 2013. The properties of logistic function and applications to neural network approximation. Journal of Computational Analysis and Applications 15, 1046–1056

2013
[29]

How to make a biological switch

Cherry, J.L., Adler, F.R., 2000. How to make a biological switch. Journal of theoretical biology 203, 117–133

2000
[30]

Mathematical modelling of molecular pathways enabling tumour cell invasion and migration

Cohen, D.P.A., Martignetti, L., Robine, S., Barillot, E., Zinovyev, A., Calzone, L., 2015. Mathematical modelling of molecular pathways enabling tumour cell invasion and migration. PLoS computational biology 11, e1004571. doi:10.1371/journal.pcbi.1004571

work page doi:10.1371/journal.pcbi.1004571 2015
[31]

Biomolecular feedback systems

Del Vecchio, D., Murray, R.M., 2015. Biomolecular feedback systems. Princeton University Press Princeton, NJ

2015
[32]

A synthetic oscillatory network of transcriptional regulators

Elowitz, M.B., Leibler, S., 2000. A synthetic oscillatory network of transcriptional regulators. Nature 403, 335

2000
[33]

Modeling the pro-inflammatory tumor microenvironment in acute lymphoblastic leukemia predicts a breakdown of hematopoietic-mesenchymal communication networks

Enciso, J., Mayani, H., Mendoza, L., Pelayo, R., 2016. Modeling the pro-inflammatory tumor microenvironment in acute lymphoblastic leukemia predicts a breakdown of hematopoietic-mesenchymal communication networks. Frontiers in physiology 7, 349. doi:10.3389/ fphys.2016.00349

work page arXiv 2016
[34]

Biochemistry 39, 11074–11083

Falcon,C.M.,Matthews,K.S.,2000.Operatordnasequencevariationenhanceshighaffinitybindingbyhingehelixmutantsoflactoserepressor protein. Biochemistry 39, 11074–11083

2000
[35]

A computational framework for optimal and model predictive control of stochastic gene regulatory networks

Faquir, H., Pájaro, M., Otero-Muras, I., 2025. A computational framework for optimal and model predictive control of stochastic gene regulatory networks. IEEE Transactions on Computational Biology and Bioinformatics

2025
[36]

Chaosinaringcircuit

Farcot,E.,Best,S.,Edwards,R.,Belgacem,I.,Xu,X.,Gill,P.,2019. Chaosinaringcircuit. Chaos:AnInterdisciplinaryJournalofNonlinear Science 29, 043103

2019
[37]

An introduction to probability theory and its applications

Feller, W., et al., 1971. An introduction to probability theory and its applications. Wiley New York

1971
[38]

Construction of a genetic toggle switch in escherichia coli

Gardner, T.S., Cantor, C.R., Collins, J.J., 2000. Construction of a genetic toggle switch in escherichia coli. Nature 403, 339–342

2000
[39]

On sharpness of error bounds for univariate approximation by single hidden layer feedforward neural networks

Goebbels, S., 2020. On sharpness of error bounds for univariate approximation by single hidden layer feedforward neural networks. Results in Mathematics 75, 109

2020
[40]

Thefive-parameterlogistic:acharacterizationandcomparisonwiththefour-parameterlogistic

Gottschalk,P.G.,Dunn,J.R.,2005. Thefive-parameterlogistic:acharacterizationandcomparisonwiththefour-parameterlogistic. Analytical biochemistry 343, 54–65

2005
[41]

Solving ordinary differential equations I: Nonstiff problems

Hairer, E., Wanner, G., Nørsett, S.P., 1993. Solving ordinary differential equations I: Nonstiff problems. Springer

1993
[42]

Corrected hill function in stochastic gene regulatory networks

Hernández-García, M.E., Velázquez-Castro, J., 2023. Corrected hill function in stochastic gene regulatory networks. arXiv preprint arXiv:2307.03057

work page arXiv 2023
[43]

Multiple regulation of the galactose operon—genetic evidence for a distinct site in the galactose operon that responds to capr gene regulation in escherichia coli k-12

Hua, S.S., Markovitz, A., 1974. Multiple regulation of the galactose operon—genetic evidence for a distinct site in the galactose operon that responds to capr gene regulation in escherichia coli k-12. Proceedings of the National Academy of Sciences 71, 507–511. doi:10.1073/ pnas.71.2.507

1974
[44]

Effects of promoter leakage on dynamics of gene expression

Huang, L., Yuan, Z., Liu, P., Zhou, T., 2015. Effects of promoter leakage on dynamics of gene expression. BMC systems biology 9, 16. doi:10.1186/s12918-015-0157-z

work page doi:10.1186/s12918-015-0157-z 2015
[45]

Mathematical modeling in systems biology: an introduction

Ingalls, B.P., 2013. Mathematical modeling in systems biology: an introduction. MIT press

2013
[46]

Genetic regulatory mechanisms in the synthesis of proteins

Jacob, F., Monod, J., 1961. Genetic regulatory mechanisms in the synthesis of proteins. Journal of molecular biology 3, 318–356

1961
[47]

Basal leakage in oscillation: Coupled transcriptional and translational control using feed-forward loops

Joanito, I., Yan, C.C.S., Chu, J.W., Wu, S.H., Hsu, C.P., 2020. Basal leakage in oscillation: Coupled transcriptional and translational control using feed-forward loops. PLOS Computational Biology 16, e1007740

2020
[48]

Metabolic stability and epigenesis in randomly constructed genetic nets

Kauffman, S.A., 1969. Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of theoretical biology 22, 437–467

1969
[49]

Stochastic gene expression modeling with hill function for switch-like gene responses

Kim, H., Gelenbe, E., 2011. Stochastic gene expression modeling with hill function for switch-like gene responses. IEEE/ACM Transactions on Computational Biology and Bioinformatics 9, 973–979

2011
[50]

Distributionofcellsizeingrowingculturesofbacteriaandtheapplicabilityofthecollins-richmondprinciple

Koch,A.,1966. Distributionofcellsizeingrowingculturesofbacteriaandtheapplicabilityofthecollins-richmondprinciple. Microbiology 45, 409–417

1966
[51]

Models of genetic networks with given properties

Kozlovska, O., Sadyrbaev, F., 2022. Models of genetic networks with given properties. WSEAS Transactions on Computer Research 10, 43–49

2022
[52]

In search of chaos in genetic systems

Kozlovska, O., Sadyrbaev, F., 2024. In search of chaos in genetic systems. Chaos Theory and Applications 6, 13–18

2024
[53]

Modeling networks of four elements

Kozlovska, O., Sadyrbaev, F., 2025. Modeling networks of four elements. Computation 13, 123

2025
[54]

Diya–auniversal light illumination platform for multiwell plate cultures

Kumar,S.,Anastassov,S.,Aoki,S.K.,Falkenstein,J.,Chang,C.H.,Frei,T.,Buchmann,P.,Argast,P.,Khammash,M.,2023. Diya–auniversal light illumination platform for multiwell plate cultures. Iscience 26

2023
[55]

Sigmoid functions: some approximation and modelling aspects

Kyurkchiev, N., Markov, S., 2015. Sigmoid functions: some approximation and modelling aspects. LAP LAMBERT Academic Publishing, Saarbrucken 4, 34

2015
[56]

Quantifyingabsoluteproteinsynthesisratesrevealsprinciplesunderlyingallocation of cellular resources

Li,G.W.,Burkhardt,D.,Gross,C.,Weissman,J.S.,2014. Quantifyingabsoluteproteinsynthesisratesrevealsprinciplesunderlyingallocation of cellular resources. Cell 157, 624–635

2014
[57]

Genetic toggle switch without cooperative binding

Lipshtat, A., Loinger, A., Balaban, N.Q., Biham, O., 2006. Genetic toggle switch without cooperative binding. Physical review letters 96, 188101

2006
[58]

Deep model predictive control of gene expression in thousands of single cells

Lugagne, J.B., Blassick, C.M., Dunlop, M.J., 2024. Deep model predictive control of gene expression in thousands of single cells. Nature Communications 15, 2148

2024
[59]

Negative auto-regulation increases the input dynamic-range of the arabinose system of escherichia coli

Madar, D., Dekel, E., Bren, A., Alon, U., 2011. Negative auto-regulation increases the input dynamic-range of the arabinose system of escherichia coli. BMC systems biology 5, 111. First Author et al.: Page 36 of 37 Short Title of the Article

2011
[60]

Quantitativemodelingoftranscription andtranslationofan all-e.colicell-freesystem

Marshall,R., Noireaux,V.,2019. Quantitativemodelingoftranscription andtranslationofan all-e.colicell-freesystem. Scientificreports9, 11980

2019
[61]

Precise modulation of transcription factor levels identifies features underlying dosage sensitivity

Naqvi,S.,Kim,S.,Hoskens,H.,Matthews,H.S.,Spritz,R.A.,Klein,O.D.,Hallgrímsson,B.,Swigut,T.,Claes,P.,Pritchard,J.K.,etal.,2023. Precise modulation of transcription factor levels identifies features underlying dosage sensitivity. Nature genetics 55, 841–851

2023
[62]

Protein degradation in escherichia coli: I

Nath, K., Koch, A.L., 1970. Protein degradation in escherichia coli: I. measurement of rapidly and slowly decaying components. Journal of Biological Chemistry 245, 2889–2900

1970
[63]

Genetic circuit design automation

Nielsen, A.A., Der, B.S., Shin, J., Vaidyanathan, P., Paralanov, V., Strychalski, E.A., Ross, D., Densmore, D., Voigt, C.A., 2016. Genetic circuit design automation. Science 352, aac7341

2016
[64]

Quality and position of the three lac operators of e

Oehler, S., Amouyal, M., Kolkhof, P., von Wilcken-Bergmann, B., Müller-Hill, B., 1994. Quality and position of the three lac operators of e. coli define efficiency of repression. The EMBO journal 13, 3348–3355

1994
[65]

Multistability in the lactose utilization network of escherichia coli

Ozbudak, E.M., Thattai, M., Lim, H.N., Shraiman, B.I., Van Oudenaarden, A., 2004. Multistability in the lactose utilization network of escherichia coli. Nature 427, 737–740

2004
[66]

Comparing different ode modelling approaches for gene regulatory networks

Polynikis, A., Hogan, S., di Bernardo, M., 2009. Comparing different ode modelling approaches for gene regulatory networks. Journal of theoretical biology 261, 511–530

2009
[67]

Tuning transcriptional regulation through signaling: a predictive theory of allosteric induction

Razo-Mejia, M., Barnes, S.L., Belliveau, N.M., Chure, G., Einav, T., Lewis, M., Phillips, R., 2018. Tuning transcriptional regulation through signaling: a predictive theory of allosteric induction. Cell systems 6, 456–469

2018
[68]

Pharmacodynamic models: parameterizing the hill equation, michaelis-menten, the logistic curve, and relationships among these models

Reeve, R., Turner, J.R., 2013. Pharmacodynamic models: parameterizing the hill equation, michaelis-menten, the logistic curve, and relationships among these models. Journal of biopharmaceutical statistics 23, 648–661

2013
[69]

On modelling of genetic regulatory networks

Sadyrbaev, F., Samuilik, I., Sengileyev, V., 2021. On modelling of genetic regulatory networks. WSEAS Transactions on Electronics 12, 73

2021
[70]

On coexistence of inhibition and activation in genetic regulatory networks, in: International Conference on Numerical Analysis and Applied Mathematics 2021, ICNAAM 2021, AIP PRESS

Sadyrbaev, F., Sengileyev, V., Silvans, A., 2023. On coexistence of inhibition and activation in genetic regulatory networks, in: International Conference on Numerical Analysis and Applied Mathematics 2021, ICNAAM 2021, AIP PRESS

2023
[71]

BMC systems biology 3, 1

Sahin,Ö.,Fröhlich,H.,Löbke,C.,Korf,U.,Burmester,S.,Majety,M.,Mattern,J.,Schupp,I.,Chaouiya,C.,Thieffry,D.,etal.,2009.Modeling erbb receptor-regulated g1/s transition to find novel targets for de novo trastuzumab resistance. BMC systems biology 3, 1

2009
[72]

Genetic engineering–construction of a network of arbitrary dimension with periodic attractor

Samuilik, I., Sadyrbaev, F., 2022. Genetic engineering–construction of a network of arbitrary dimension with periodic attractor. Vibroengi- neering Procedia 46, 67–72

2022
[73]

Mathematicalmodelingofthree-dimensionalgeneticregulatorynetworksusinglogisticand gompertz functions

Samuilik,I.,Sadyrbaev,F.,Ogorelova,D.,2022. Mathematicalmodelingofthree-dimensionalgeneticregulatorynetworksusinglogisticand gompertz functions. WSEAS Transactions on systems and control 17, 101107

2022
[74]

Ontheuseofthehillfunctionsinmathematicalmodelsofgeneregulatorynetworks

Santillán,M.,2008. Ontheuseofthehillfunctionsinmathematicalmodelsofgeneregulatorynetworks. MathematicalModellingofNatural Phenomena 3, 85–97

2008
[75]

Sawlekar, R., Montefusco, F., Kulkarni, V., Bates, D.G., 2015. Biomolecular implementation of a quasi sliding mode feedback controller basedondnastranddisplacementreactions,in:201537thAnnualInternationalConferenceoftheIEEEEngineeringinMedicineandBiology Society (EMBC), IEEE. pp. 949–952

2015
[76]

Detailed map of a cis-regulatory input function

Setty, Y., Mayo, A.E., Surette, M.G., Alon, U., 2003. Detailed map of a cis-regulatory input function. Proceedings of the National Academy of Sciences 100, 7702–7707

2003
[77]

Revealing gene regulation-based neural network computing in bacteria

Somathilaka, S.S., Balasubramaniam, S., Martins, D.P., Li, X., 2023. Revealing gene regulation-based neural network computing in bacteria. Biophysical Reports 3

2023
[78]

Logical model specification aided by model-checking techniques: application to the mammalian cell cycle regulation

Traynard, P., Fauré, A., Fages, F., Thieffry, D., 2016. Logical model specification aided by model-checking techniques: application to the mammalian cell cycle regulation. Bioinformatics (Oxford, England) 32, i772–i780. doi:10.1093/bioinformatics/btw457

work page doi:10.1093/bioinformatics/btw457 2016
[79]

A comprehensive approach to the molecular determinants of lifespan using a boolean model of geroconversion

Verlingue, L., Dugourd, A., Stoll, G., Barillot, E., Calzone, L., Londoño-Vallejo, A., 2016. A comprehensive approach to the molecular determinants of lifespan using a boolean model of geroconversion. Aging cell 15, 1018–1026. doi:10.1111/acel.12504

work page doi:10.1111/acel.12504 2016
[80]

The galactose regulon of escherichia coli

Weickert, M.J., Adhya, S., 1993. The galactose regulon of escherichia coli. Molecular microbiology 10, 245–251

1993

Showing first 80 references.