pith. sign in

arxiv: 2604.05888 · v1 · submitted 2026-04-07 · 🧮 math.DS

Mathematical modeling and analysis of the Notch-Delta pathway

Angela Stevens , Nicola Vassena This is my paper

Pith reviewed 2026-05-10 18:50 UTC · model grok-4.3

classification 🧮 math.DS
keywords Notch-Delta pathwaysymmetry-induced bifurcationsreaction networksstoichiometric modelscell differentiationODE systemssingular Jacobianminimal models
0
0 comments X

The pith

The Notch-Delta pathway can break symmetry between neighboring cells through bifurcations that depend only on network structure.

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

The authors build reaction-network models of the Notch-Delta pathway between two adjacent cells using only the stoichiometry and the species each reaction depends on. From these they derive ODE systems and examine the conditions under which a symmetric steady state can lose stability via a symmetry-induced bifurcation. The analysis is carried out symbolically and remains valid for broad families of rate laws because the steady-state fluxes and the derivatives of the rates at those states are treated as independent parameters. This structure-based approach lets them test which interactions in the pathway are necessary for one cell to adopt a high-Notch state and its neighbor a high-Delta state.

Core claim

Using stoichiometric reaction networks for the Notch-Delta pathway, the associated ODE systems admit a singular Jacobian at the symmetric equilibrium for a wide class of kinetics. The singularity is detected symbolically once the steady-state fluxes and the first derivatives of the reaction rates are parametrized independently. Abstract minimal models derived from the full network then isolate which interactions are required for the symmetry-breaking bifurcation that produces distinct cell fates.

What carries the argument

The symbolic test for a singular Jacobian at the symmetric steady state of the two-cell ODE system derived from the reaction-network stoichiometry.

If this is right

  • The network structure itself is sufficient to permit lateral inhibition or cell differentiation without requiring specially tuned rate constants.
  • Minimal subnetworks extracted from the full pathway can be shown to retain or lose the capacity for symmetry-induced bifurcations.
  • The same symbolic Jacobian test applies to any parameter-rich kinetics whose fluxes and derivatives remain independent at steady state.
  • Certain interactions identified in the minimal models are necessary for the bifurcation while others can be removed without destroying it.

Where Pith is reading between the lines

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

  • The same stoichiometric test could be applied to other juxtacrine signaling systems to see which ones are structurally capable of symmetry breaking.
  • Experimental perturbation of the interactions flagged as essential by the minimal models should block or restore cell-fate divergence in vivo.
  • Extending the two-cell network to a lattice of cells would allow the same Jacobian analysis to predict patterns such as stripes or spots.

Load-bearing premise

The steady-state fluxes and the first derivatives of the reaction rates at those states can be parametrized independently.

What would settle it

A concrete kinetic model of the Notch-Delta network in which the Jacobian matrix at the symmetric steady state remains nonsingular for all choices of the independent flux and derivative parameters would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.05888 by Angela Stevens, Nicola Vassena.

Figure 1
Figure 1. Figure 1: Schematic representation of the biological mechanisms taken into account in our mathematical models, on the left BI and the cis-reactions BII, on the right the ligand-activation variant BIII. Not all interactions are depicted symmetrically, for visual clarity, but they are to be understood as occurring symmetrically in both cells. Letters refer to the biochemical explanation in the text. For the complete h… view at source ↗
Figure 2
Figure 2. Figure 2: Instability motifs in the cis-model BI+BII (rows 1, 2, 3; motifs (3.5), (3.6), and (3.7)) and in the ligand-activation model BIII (row 4; motif (3.10)). The cis-reactions 14 − 15, and 24 − 25, are marked by dashed arrows. The left and right columns display the symmetric variants. The motifs in rows 1, 2 share the same topological structure, differing only by the use of NIj in the first and Tj in the second… view at source ↗
Figure 3
Figure 3. Figure 3: The steady states of (4.11) are plotted as a function of β using Python. Continuous lines signify stable steady states, and dashed lines unstable ones. The boundary steady states EB1 = 0 and EB2 = 1 are unstable, irrespective of the value of β. In turn, at β = 2, the reference steady state Ek = 0.5 loses stability and undergoes a supercritical pitchfork bifurcation. For β > 2 bistability occurs. whose non-… view at source ↗
Figure 4
Figure 4. Figure 4: Networks NonAut-I-2 (left) and NonAut-II-2 (right), with instability motifs. Remark 4.8. In contrast to NonAut-I-η, the model NonAut-II-η has the capacity for differentiation for all η. However, for condition (4.29) to hold for NonAut-II-1 at kinetic symmetry at a homogeneous steady-state, it is necessary that the functions r1, r3 are not symmetric, i.e., r(x, y) ̸= r(y, x). This is a crucial restriction, … view at source ↗
Figure 5
Figure 5. Figure 5: Top row: models MI (top left), MIII (top center), NonAut-II-2 (top right). Bottom row: models MIIIb (bottom left), and NonAut-II-1 (bottom right). Black squares denote Λ1 and grey squares Λ2. The white rectangle denotes NI1 in MIIIb, and I2 in NonAut-II-η. As in [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
read the original abstract

In this paper mathematical models for the evolutionary conserved Notch-Delta pathway are developed and analyzed in order to better understand how two neighboring biological cells can become different. We pursue a structure-based stoichiometric type of approach, such that no specific reaction kinetics have to be defined. Only their dependencies on the relevant species participating in the model network are taken into account. Reaction networks and their related systems of ODEs are presented and analyzed with respect to their capacity for symmetry-induced bifurcations. The possibility to obtain a singular Jacobian is analyzed symbolically. This approach is valid for parameter-rich kinetics, where the parametrization of the steady-state fluxes and of the first derivatives of the reaction rates evaluated at the steady state are independent. In this context, also with the help of abstract minimal models, we could mathematically identify some of the Notch pathway's features being more relevant than others.

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 / 1 minor

Summary. The paper develops structure-based stoichiometric models of the Notch-Delta pathway without specifying explicit reaction kinetics, instead using only species dependencies to construct reaction networks and associated ODE systems. These are analyzed for symmetry-induced bifurcations and the possibility of singular Jacobians, with the latter performed symbolically under the assumption that steady-state flux parametrizations are independent of the first derivatives of reaction rates evaluated at steady state (valid for parameter-rich kinetics). Abstract minimal models are employed to identify which structural features of the Notch pathway are more relevant for neighboring cells becoming different.

Significance. If the independence assumption holds and the symbolic analysis is rigorous, the structure-based approach offers a parameter-independent route to rank the relevance of network features in Notch-Delta signaling, which could aid understanding of symmetry breaking in cell differentiation. The avoidance of specific kinetics is a methodological strength that aligns with the goal of identifying broadly applicable structural determinants.

major comments (2)
  1. [Abstract] Abstract: The central claim that abstract minimal models and structure-based analysis can identify more relevant Notch pathway features rests on the ability to symbolically detect a singular Jacobian. This detection is conditioned on the parametrization of steady-state fluxes being independent from the first derivatives of the reaction rates at steady state. No derivation or argument is supplied showing that the actual conservation relations and rate-law dependencies in the Notch-Delta network permit such independent variation; if the quantities are coupled once explicit kinetics are substituted, the singularity condition may describe no realizable parameter point.
  2. [Abstract] Abstract and subsequent analysis sections: The manuscript states that the symbolic approach is valid for parameter-rich kinetics but does not provide the explicit minimal models or the full symbolic derivations of the Jacobian singularity condition. Without these, it is not possible to verify whether the identified relevant features follow rigorously from the stated network structure rather than from the independence assumption alone.
minor comments (1)
  1. [Abstract] The abstract would benefit from a brief indication of the specific sections or equations where the abstract minimal models are constructed and where the symbolic Jacobian analysis is carried out.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our paper. The comments highlight important aspects regarding the justification of our assumptions and the presentation of our methods. We respond to each major comment below and indicate the revisions we will make to address them.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that abstract minimal models and structure-based analysis can identify more relevant Notch pathway features rests on the ability to symbolically detect a singular Jacobian. This detection is conditioned on the parametrization of steady-state fluxes being independent from the first derivatives of the reaction rates at steady state. No derivation or argument is supplied showing that the actual conservation relations and rate-law dependencies in the Notch-Delta network permit such independent variation; if the quantities are coupled once explicit kinetics are substituted, the singularity condition may describe no realizable parameter point.

    Authors: We acknowledge the validity of this concern. The independence assumption is posited for parameter-rich kinetics in general, but we did not provide a specific argument tailored to the conservation relations in the Notch-Delta models. To strengthen the manuscript, we will add a new subsection deriving that the structure of our stoichiometric models allows independent parametrization of steady-state fluxes and rate derivatives at equilibrium, using the degrees of freedom in parameter-rich rate laws. This will demonstrate that the singularity condition can be realized. revision: yes

  2. Referee: [Abstract] Abstract and subsequent analysis sections: The manuscript states that the symbolic approach is valid for parameter-rich kinetics but does not provide the explicit minimal models or the full symbolic derivations of the Jacobian singularity condition. Without these, it is not possible to verify whether the identified relevant features follow rigorously from the stated network structure rather than from the independence assumption alone.

    Authors: We agree that including the explicit minimal models and the symbolic derivations would enhance the rigor and verifiability of our results. Although the models are described in the main text and derivations are outlined symbolically, we will revise the manuscript to present the key abstract minimal models explicitly and provide the step-by-step symbolic computation of the Jacobian singularity condition in the main body or a dedicated appendix section. This will clarify that the identification of relevant features stems from the network structure. revision: yes

Circularity Check

0 steps flagged

No circularity: structure-based analysis remains independent of fitted inputs or self-referential reductions

full rationale

The paper develops reaction networks and ODEs via a stoichiometric structure-based approach that avoids specifying particular kinetics, instead using only species dependencies. It analyzes symmetry-induced bifurcations and the symbolic conditions for a singular Jacobian, explicitly conditioning the latter on the independence of steady-state flux parametrizations from rate derivatives at equilibrium for parameter-rich kinetics. Abstract minimal models are then invoked to rank feature relevance. No quoted step equates a derived prediction or identification to an input quantity by construction, nor does any load-bearing claim reduce to a self-citation, fitted parameter, or ansatz smuggled from prior work. The independence statement functions as a stated validity scope rather than a derived or tautological result, leaving the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the independence of steady-state flux parametrization and rate derivatives at equilibrium, plus standard assumptions of ODE existence and local stability analysis for reaction networks.

axioms (2)
  • domain assumption The parametrization of the steady-state fluxes and of the first derivatives of the reaction rates evaluated at the steady state are independent.
    Invoked to enable symbolic analysis of the singular Jacobian in parameter-rich kinetics.
  • standard math Reaction networks admit systems of ODEs whose steady states can be analyzed for symmetry-induced bifurcations.
    Background assumption of dynamical systems theory applied to the stoichiometric models.

pith-pipeline@v0.9.0 · 5434 in / 1285 out tokens · 32658 ms · 2026-05-10T18:50:36.623600+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    DOI 10.48550/arXiv.2508.15273, 2508.15273

    [BSV25] Alexander Blokhuis, Peter F Stadler, and Nicola Vassena. Stoichiometric recipes for periodic oscillations in reaction networks.arXiv preprint arXiv:2508.15273,

    work page arXiv

  2. [2]

    DOI 10.48550/ARXIV.2511.14431

    39 [GGSV25] Richard Golnik, Thomas Gatter, Peter F Stadler, and Nicola Vassena. BiRNe: Symbolic bifurcation analysis of reaction networks with Python.arXiv preprint arXiv:2511.14431,

    work page arXiv

  3. [3]

    [MR12] Stefan M¨ uller and Georg Regensburger. Generalized mass action systems: Com- plex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces.SIAM Journal on Applied Mathematics, 72(6):1926–1947,

    work page 1926