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Modelling Population-Level Hes1 Dynamics: Insights from a Multi-Framework Approach

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A minimal spatial model of the Hes1-Notch network produces transient oscillations followed by a stable salt-and-pepper cell-fate pattern that survives intrinsic noise.

desk verdict A genuinely useful multi-framework Hes1-Notch modeling paper whose reduced-model stability analysis is sound, but the full-model patterned-stability proof has a real gap and the parameterization is partly fitted. read the letter →

arxiv 2411.09721 v2 pith:AHNAWUCK submitted 2024-11-14 q-bio.MN math.DSq-bio.CB

Gesina Menz , Stefan Engblom This is my paper · ORCID
classification q-bio.MNmath.DSq-bio.CB MSC 92-1092B2592C1534A3360J2034C6034F10
keywords Hes1Notchsignalinglateralinhibitionpatternformationgeneticoscillatorquasi-steady-statereductionreaction-diffusionmasterequationneuralprogenitordifferentiation
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

Hes1 is a transcription factor whose levels oscillate in neural progenitors, and the final high/low pattern of Hes1 expression is tied to whether a cell becomes a neuron or a glial cell. This paper tries to establish that a minimal spatial model of the Hes1-Notch network—five molecular species per cell on a hexagonal grid—can reproduce both the damped oscillations and the final salt-and-pepper pattern without adding external regulation. The authors reduce the five-species equations to two-dimensional and scalar systems by assuming some molecules reach steady state quickly, then prove that the homogeneous state is unstable under explicit conditions and that a non-homogeneous patterned state exists and is stable. They also show in a stochastic reaction-diffusion master equation version that the pattern survives intrinsic noise, with a patterning coefficient of 0.50 (95% CI 0.43–0.57) at a physiological cell volume of 50 µm3. If this is right, the oscillatory-then-sustained expression modes seen in development may be intrinsic to the Hes1-Notch gene regulatory network itself.

What carries the argument

The object that carries the argument is a pair of decreasing Hill-type functions, $f(x)=1/(a+x^k)$ and $g(x)=1/(1+b x^h)$, which arise after quasi-steady-state reductions and represent Hes1-mediated repression and Notch-mediated activation of Hes1 mRNA. The stability analysis turns on the derivative balance $f(\bar{x}_0)g'(\bar{x}_0)-f'(\bar{x}_0)g(\bar{x}_0)<-1$: it decides whether a perturbation from the homogeneous state grows or decays, and the same balance appears in the full five-species model as condition (3.15) through a determinant-preservation lemma for the state reduction. For the spatial pattern, Fourier analysis on the hexagonal lattice selects the mode with period three lattice steps, and a three-cell model with coupling matrix $W_3$ serves as the minimal system that reproduces the full grid's bifurcation structure.

What would settle it

Measure the model's parameters (activation rates, half-lives, Hill constants, and cell-cell coupling strengths) from mouse neural progenitor cells, insert them into condition (3.15), and check whether the left-hand side is below $-1$; if it is not, the predicted loss of homogeneity does not occur in the model with measured parameters.

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Extended reading notes

Core claim

The paper's central claim is that the Hes1-Notch signalling network, written as a five-variable ODE per cell coupled over a regular hexagonal grid, settles from transient oscillations into a stable static pattern in which neighbouring cells express high and low Hes1 alternately. Using quasi-steady-state reductions, the paper derives reduced equations of the form $\dot{x}=\langle y_{\mathrm{in}}\rangle f(x)-x$, $\dot{y}=v(g(x)-y)$ and a scalar form $\dot{x}=\langle g(x_{\mathrm{in}})\rangle f(x)-x$, where $f$ and $g$ are decreasing Hill-type functions. Propositions 3.2, 3.3, 3.4, 3.6, and 3.7 show that the homogeneous steady state is unstable exactly when $f(\bar{x}_0)g'(\bar{x}_0)-f'(\bar{x}_0)g(\bar{x}_0)<-1$ (equivalently condition (3.15) in the full model), and that whenever a non-homogeneous solution exists it is stable. On a hexagonal lattice the most unstable Fourier mode has period three lattice steps, matching the observed checkerboard pattern, and a three-cell reduction with coupling matrix $W_3$ reproduces the bifurcation behaviour. In the stochastic RDME formulation, the patterned state remains close to perfect even at high intrinsic noise, with patterning coefficient $\hat{p}=0.50$ (95% CI 0.43–0.57) at 50 µm3.

Load-bearing premise

The parameters controlling activation, repression, and Hill coefficients are fitted or chosen ad hoc—not directly measured—so the predicted instability and patterning may not hold for real Hes1 systems if those values are wrong.

Editorial extensions

If this is right

  • The explicit instability conditions (3.2), (3.15), and (3.19) give parameter checks for predicting when a uniform Hes1 population will develop into a high/low patterned state.
  • The same reduction framework, with identical parameters $a$ and $b$ but different time-scale parameter $v$, yields seven reduced systems, so the timing of fate decision, not the steady-state pattern, is what changes between reductions.
  • The stochastic RDME model preserves checkerboard patterning at cell volumes down to 1 µm3, and at 50 µm3 the patterning coefficient is 0.50 (95% CI 0.43–0.57), close to perfect alternation.
  • Damped oscillations followed by stable high/low expression can arise from the Hes1-Notch network alone, suggesting the two expression modes need not be imposed by external signals.
  • If Notch activation strength is weakened, the model predicts the patterned state is lost through a bifurcation, which could be tested by titrating Notch signalling in a neural progenitor population.

Reading between the lines

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

  • A direct test of the model's core mechanism would be to measure the patterning coefficient in living neural progenitors with lineage tracing; if the observed high-high coupling is not near 1/2, the specific salt-and-pepper prediction would fail even if the instability condition holds.
  • The same quasi-steady-state reduction strategy could transfer to other lateral-inhibition systems, such as Notch-Delta models with different Hill exponents, to predict whether they also exhibit a three-cell minimum pattern.
  • Because the parameters $k=1$ and $h=4$ were chosen as the smallest values producing oscillations, the framework suggests a parameter scan of steeper Hill coefficients (or added delay) could make oscillations less damped without changing the patterning result.
  • If real neural progenitors show noise-robust patterning, the RDME result implies that stochasticity acts on the timing of fate decision rather than on the pattern itself, a distinction that single-cell tracking could resolve.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper develops an ODE model of the Hes1-Delta/Notch gene regulatory network on a hexagonal cell grid, with parameters taken from the literature and partly fitted. It derives reduced two-dimensional and scalar models via quasi-steady-state assumptions, analyzes stability of homogeneous and patterned states in a two-cell periodic setting, extends the analysis to the full model, studies patterning on hexagonal tilings, and compares deterministic results with an RDME stochastic model. The main claims are: a unique homogeneous steady state; instability conditions (3.2), (3.15), and (3.19); existence and stability of a non-homogeneous patterned state in the reduced and full models; and preservation of patterning in the RDME model with a patterning coefficient of 0.50 (95% CI 0.43–0.57) at 50 µm³ cell volume.

Significance. If the analytical results were fully established, the paper would provide a useful bridge between deterministic grid ODE models and stochastic RDME models of a well-studied developmental signalling system. Notable strengths are the candid discussion of parameter limitations (Section 4), the reproducible URDME workflow (Section 4.1), and the combination of multiple modelling frameworks. The numerical bifurcation diagrams and RDME simulations give credible qualitative support for the proposed patterning mechanism. However, the central analytical proof of full-model patterned-state stability is incomplete, and the parameters governing the bifurcation are partly fitted to produce the observed behaviours, so the predictive content is currently limited.

major comments (3)
  1. [§3.2, Propositions 3.6 and 3.7, Eq. (3.15), Lemma 3.5] The proofs of Propositions 3.6 and 3.7 assert that positivity of all non-constant coefficients of the characteristic polynomial, together with the sign of the constant term, determines stability. This is not valid for polynomials of degree at least three: unstable complex-conjugate root pairs can occur even when all coefficients are positive and the constant term is positive. Consequently, the "if and only if" statement in Proposition 3.6 and the conclusion in Proposition 3.7 that "the stability condition (3.15) again controls the stability" are not established by the arguments given. In Proposition 3.7 specifically, condition (3.15) is derived for the homogeneous state P0, while the constant term of the characteristic polynomial for the patterned state is evaluated at P1 and P2; no argument connects the sign of that constant term to (3.15). The numerical bifurcation diagrams (Figs. 3.2 and 3.5) support the intended conclusion, but the analytical proof of full-model stability of the patterned state, which underpins the subsequent RDME patterning claim, is incomplete. I recommend either verifying the full Routh–Hurwitz conditions for the specific polynomial structure, or computing the relevant eigenvalues numerically for the reported parameter ranges, and then adjusting the analytical claims accordingly.
  2. [§2.2, Table 2.1, Appendix A, Section 4] The model behaviour is parameterized to produce the desired outputs: the activation rates αi are fitted to PaxDB concentrations and to give the desired dynamics (caption of Table 2.1), KM and Kn are chosen to match overall system behaviour, and k = 1 and h = 4 are selected as the minimum values yielding oscillations. As a result, the instability conditions (3.2), (3.15), and (3.19), and the existence of the patterned state are not independent predictions for the Hes1 system but rather consequences of the chosen parameter values. The paper acknowledges this limitation in Section 4, but the central claim that the patterning is stable under intrinsic noise (Section 3.4) would be considerably strengthened by a sensitivity analysis over KM, Kn, k, and h, which are held fixed without perturbation. This is a scientific rather than mathematical gap; it does not invalidate the analysis conditional on the chosen parameters.
  3. [§3.4, Fig. 3.8] The 95% confidence interval for the patterning coefficient p is obtained by treating high–high couplings as independent Bernoulli trials. Since neighbouring pairs share cells and the spatial pattern is strongly correlated, the effective number of independent samples is smaller than the number of counted couplings, so the reported interval (0.43, 0.57) may be overconfident. A block bootstrap or a spatial resampling procedure would give a more defensible uncertainty estimate. This affects the quantitative statement of patterning coefficient 0.50, though not the qualitative conclusion that patterning is robust to noise for the modelled parameter regime.
minor comments (6)
  1. [§3.1, Proposition 3.1] The uniqueness of the homogeneous steady state for the full system (2.2) is asserted "by extension" without a proof; since the full model involves five species in each cell, a brief argument or a reference would make this step transparent.
  2. [§3.1, Eq. (3.7)] The statement that γ′2(¯x1) = γ′2(¯x2) < 1 is justified only "by inspection" and by the graphical illustration in Fig. 3.1; an explicit inequality for these derivatives would be more convincing.
  3. [§3.3, Proposition 3.9, Eq. (3.25)] The proof of Proposition 3.9 relies on a "graphical motivation" for the double-root condition, and the displayed formula (3.25) would benefit from the intermediate differentiation steps being shown, since the expression is not self-evident.
  4. [§2.3, Eq. (2.11)] The pseudo-species Din is introduced as a diffusing signal but no degradation or decay term is specified; clarify whether Din has a finite lifetime and how the rate αN qkl is derived from the cell-contact signalling process.
  5. [§4.1] The reproducibility statement refers to the "Hes1 directory" in the DLCM workflow but does not provide a DOI, version number, or exact URL for the workflow; adding these would make the results reproducible in practice.
  6. [Table 2.1] The entries for µD and µN contain garbled typesetting (e.g., "log 2/50 × 5 log(2) /(45.3, 55.2) × 5"); the intended expression appears to be 5·log(2)/(45.3–55.2) min⁻¹ and should be typeset accordingly.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: oscillatory dynamics are fit inputs, while the patterning-stability derivation is self-contained.

  1. fitted input called prediction [Section 2.2 and Table 2.1; Discussion, §4]
    "Values of αi are chosen to give the desired behaviour of constituents relative to each other [24] ... we choose the Hill-function dissociation constants KM and Kn to match the overall system behaviour, as these primarily influence the oscillation period and number of oscillations. ... We require both k, h∈ N+ and choose k = 1 and h = 4 as these are the minimum values which we have found are necessary to realistically capture oscillations. ... [O]ur results suggest that these distinct expression modes may be intrinsic to the GRN, and thus independent of external regulation."

    The activation rates αi, the dissociation constants KM and Kn, and the Hill coefficients k and h are explicitly fitted so that the model reproduces the desired oscillatory behavior and concentration scales. The Discussion then treats the resulting oscillatory and sustained expression modes as a model-derived insight, suggesting they are intrinsic to the GRN. This is a fitted input presented as a result: the model's ability to oscillate is a restatement of the fitting choices, not an independent prediction. The patterning-stability theorems (Propositions 3.1–3.8) are genuine derivations from the fitted equations, and the RDME patterning coefficient is simulated rather than fit, so the circularity is partial and does not infect the central stability analysis.

full rationale

The stability analysis of the homogeneous and non-homogeneous states (Propositions 3.1–3.8) is a genuine mathematical derivation from the model equations: conditions (3.2), (3.15), and (3.19) are obtained by linearization and characteristic-polynomial analysis, not by imposing the conclusion. The RDME robustness study is an independent simulation check using the same fitted parameters, with the patterning coefficient estimated from the stochastic model rather than used as a fit target. However, the paper's oscillation behavior is an input constraint: αi, KM, Kn, k, and h are explicitly chosen to produce the observed oscillatory dynamics, and the Discussion then presents the oscillatory versus sustained expression modes as a model result suggesting they are intrinsic to the GRN. This part is partly circular because the fitted parameters already contain the target behavior. The self-citations to URDME ([5], [12]) are software/framework citations and are not load-bearing for the analytical claims. A separate correctness concern, not circularity, is that Proposition 3.7 asserts the degree-10 patterned-state characteristic polynomial is stable when coefficients are positive and (3.15) holds, but positivity of non-constant coefficients is not sufficient for stability of a degree-10 polynomial, and the proof does not establish equivalence between p(0)<0 at the patterned state and condition (3.15).

Assumptions & free parameters 11 free parameters · 5 assumptions · 1 invented entities

The central stability results are derived from a model whose parameters are partly fitted to concentration data and partly chosen ad hoc. The free parameters listed above are the main inputs that determine the qualitative behavior (oscillations and patterning). The axioms include modeling assumptions (QSS, Hill forms, grid extension) and standard mathematics. The only invented entity is the computational pseudo-species Din, which is not a biological claim.

free parameters (11)
  • αD = 0.018 /min (0.016, 0.021)
    Fitted to match target Dll1 concentration from PaxDB and desired oscillation behavior (Section 2.2, Appendix A).
  • αN = 6.0 /min (5.3, 6.7)
    Fitted to match target Notch concentration and oscillation behavior (Section 2.2, Appendix A).
  • αM = 0.017 /min (0.016, 0.019)
    Fitted to match Hes1 mRNA concentration and oscillation behavior (Section 2.2, Appendix A).
  • αP = 0.14 /min (0.12, 0.16)
    Fitted to match Hes1 protein concentration and oscillation behavior (Section 2.2, Appendix A).
  • αn = 0.0049 µM/min (0.0043, 0.0054)
    Fitted to match Ngn2 concentration and oscillation behavior (Section 2.2, Appendix A).
  • KM = 0.050 µM
    Chosen to match overall system behavior (oscillation period and number), not measured (Section 2.2).
  • Kn = 0.030 µM
    Chosen to match overall system behavior (Section 2.2).
  • k = 1
    Chosen as the minimum Hill coefficient needed to realistically capture oscillations (Section 2.2).
  • h = 4
    Chosen as the minimum Hill coefficient needed to realistically capture oscillations (Section 2.2).
  • Bound fraction for D and N = 80%
    Ad hoc assumption that 80% of Dll1 and Notch are bound and inactive, leading to a 5-fold increase in degradation rates (Section 2.2, Table 2.1).
  • Concentration scaling uncertainty = 5% relative
    Ad hoc uncertainty applied to PaxDB concentrations when fitting αi (Appendix A).
assumptions (5)
  • domain assumption Quasi-steady-state assumptions for N, P, n (and later y in scalar reduction)
    Used to reduce (2.2) to (2.3)-(2.5); assumes these species equilibrate quickly, not validated in the paper (Appendix B).
  • domain assumption Hill-function forms for repression of M and n by P
    Phenomenological Hill functions with exponents k=1 and h=4 are assumed for the GRN (Section 2.2).
  • domain assumption Two-cell periodic analysis extends to the full grid
    Propositions 3.2-3.4 are proved for two cells; the extension to hexagonal grids is assumed via the most unstable Fourier mode (Section 3.3).
  • standard math Block determinant Lemma 3.5
    Uses the Schur complement determinant identity for the full Jacobian; standard linear algebra.
  • standard math Monotonicity of f and g
    Used to establish the unique homogeneous steady state in Proposition 3.1.
invented entities (1)
  • Din (diffusing Dll1 signal pseudo-species)
    purpose: Models the direct cell-to-cell Dll1 signal that converts to Notch in neighboring cells in the RDME model (reactions (2.11)).
    A computational abstraction; no direct biological counterpart is claimed, and it is not a measured entity.

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Cite this review

Pith. "Pith review of Modelling Population-Level Hes1 Dynamics: Insights from a Multi-Framework Approach." pith.science (2026). https://pith.science/paper/AHNAWUCK

@misc{pith2026241109721,
  author       = {Pith},
  title        = {Pith review of: Modelling Population-Level Hes1 Dynamics: Insights from a Multi-Framework Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHNAWUCK}},
  note         = {Machine review of arXiv:2411.09721}
}
read the original abstract

Mathematical models of living cells have been successively refined with advancements in experimental techniques. A main concern is striking a balance between modelling power and the tractability of the associated mathematical analysis. In this work we model the dynamics for the transcription factor Hairy and enhancer of split-1 (Hes1), whose expression oscillates during neural development, and which critically enables stable fate decision in the embryonic brain. We design, parametrise, and analyse a detailed spatial model using ordinary differential equations (ODEs) over a grid capturing both transient oscillatory behaviour and fate decision on a population-level. We also investigate the relationship between this ODE model and a more realistic grid-based model involving intrinsic noise using mostly directly biologically motivated parameters. While we focus specifically on Hes1 in neural development, the approach of linking deterministic and stochastic grid-based models shows promise in modelling various biological processes taking place in a cell population. In this context, our work stresses the importance of the interpretability of complex computational models into a framework which is amenable to mathematical analysis.

Figures

Figures reproduced from arXiv: 2411.09721 by the authors.

Figure 2.1
Figure 2.1. Left: Representation of neurons (orange), glial cells (blue) and undifferentiated cells (pink) in a developing brain. Right: Schematics of the Hes1 negative feedback loop in two neighbouring cells. The same interactions occur in every cell throughout the neural progenitor cell population between all neighbouring cells. All arrows ending with an arrowhead denote an activation or creation of a constituent while arrows… view at source ↗
Figure 2.2
Figure 2.2. Dynamics averaged over all cells on a 20-by-20 grid of hexagonal cells when [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. (a)–(d): spatial dynamics of Hes1 mRNA in our proposed grid ODE model (2.2) where blue cells are above the mean concentration before fate decision and orange cells are below this threshold. (e)–(h): Hes1 mRNA in the reduced model (2.3) on the same grid. (i): the average Hes1 mRNA (solid line: full ODE model; dashed line: reduced model) over time calculated separately over all cells which show high or low Hes1 concen… view at source ↗
Figures from the paper (10 more)
Figure 2.4
Figure 2.4. Figure 2.4: The schematics as implemented on a hexagonal grid (solid lines) using the RDME model. The dashed lines show the triangulation on which the hexagonal grid is built in the URDME framework. in the lth voxel, and where (Π· , Π′ · ) is an appropriately extended set of ind…
Figure 2.5
Figure 2.5. Figure 2.5: (a)–(d): spatial dynamics of Hes1 mRNA in our RDME model (2.9) and (2.11) choosing the volume of each voxel to be 1µm3 , representing a rather high noise levels, and using the same colour scheme as in [PITH_FULL_IMAGE:figures/full_fig_p011_2_5.png]
Figure 3.1
Figure 3.1. Figure 3.1: Fix point arguments. Left: the unique homogeneous stationary state is the fix point ¯x0 = φ(¯x0). Right: if γ ′ 2 (¯x0) > 1, then there are cyclic (non-homogeneous) solutions x¯1 < x¯0 < x¯2. Proposition 3.3. Under condition (3.2) there exists a non-homogeneous stati…
Figure 3.2
Figure 3.2. Figure 3.2: The non-homogeneous and ho￾mogeneous stationary states of the Hes1 protein P (log-scale), respectively, as a function of a scaling s, which acts upon the parameter αN , by scaling αN 7→ s −1αN . The homogeneous solution always exists, but is unstable to the left of t…
Figure 3.3
Figure 3.3. Figure 3.3: A regularly periodic pattern on a hexagonal grid with period 3 in each lattice direction. The labelling scheme shown follows [10]. The red squares highlight two vertices which make the pattern non-transitive: the left borders 2 white and one black cell, while the rig…
Figure 3.4
Figure 3.4. Figure 3.4: The three patterns on a regular hexagonal tiling which show vertex transitive [PITH_FULL_IMAGE:figures/full_fig_p017_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Stationary solutions under vanishing feedback. [PITH_FULL_IMAGE:figures/full_fig_p020_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Phase-plot of the three-cell prob￾lem in the plane which contains all three stationary points. Circle: stable non￾homogeneous solution, triangles: unstable solutions with red the homogeneous one. Level curves according to the Euclidean norm of the right-hand side and…
Figure 3.7
Figure 3.7. Figure 3.7: Same scaling as in Fig [PITH_FULL_IMAGE:figures/full_fig_p021_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: A measure of connectivity between high and low protein cells (per￾fect patterning corresponds to pattern￾ing coefficient p = 1/2) for a sequence of volumes in the RDME model (2.9). The numerical problem is treated as a statistical estimation problem for which the con…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.