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

TL;DR

This work tackles how population-level Hes1 dynamics arise from cellular signaling, oscillations, and fate decisions in neural development. It couples a deterministic grid-ODE representation of the Hes1-Notch network with a spatial stochastic RDME to capture intrinsic noise across a hexagonal cell lattice, enabling analysis of both oscillatory and patterning behaviour. The authors show a unique homogeneous steady state in the reduced system, which becomes unstable and yields stable non-homogeneous patterns; these patterns persist in the full model and are reproducible under noise in the RDME framework, with hexagonal tilings supporting vertex-transitive checkerboard-like states. The study demonstrates a principled linkage between deterministic and stochastic population-level descriptions, offering a tractable yet biologically informative framework that can be extended to other developmental GRNs and to assess robustness to intrinsic noise.

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 (7)

• 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 ending with a vertical line denote a repression. All constituents are also degraded (not shown).
• Figure 2.2: Dynamics averaged over all cells on a $20$-by-$20$ grid of hexagonal cells when starting from random initial data. The two dashed vertical lines indicate the offset between Hes1 mRNA and Hes1 protein expression levels which has been shown previously in Hirata2002OscillatoryLoop. The offset between the Hes1 mRNA and Hes1 protein oscillations between the two markers as shown is approximately $30$ minutes. The vertical dotted line shows that we approximately capture the inverse oscillations between the Hes1 protein, and Dll1 and Ngn2 Shimojo2011DynamicCells.
• Figure 2.3: (a)--(d): spatial dynamics of Hes1 mRNA in our proposed grid ODE model \ref{['eq:ode']} 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 \ref{['eq:red_type1']} 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 concentrations after fate decision with blue and orange denoting high and low expression, respectively. The vertical lines denote the times at which the spatial dynamics are shown in the top and middle rows. All simulations shown are on a $20 \times 20$ grid with zero boundary conditions. Initial conditions are uniform random values scaled to the required concentrations as given in Appendix \ref{['app:parameters']}. Note that by our parameterisation, we find our results in concentrations.
• Figure 2.4: (a)--(d): spatial dynamics of Hes1 mRNA in our RDME model \ref{['eq:RDME1']} and \ref{['eq:RDME2']} choosing the volume of each voxel to be $1 \mu m^3$, representing a rather high noise levels, and using the same colour scheme as in Fig. \ref{['fig:ode_panel']}. (e)--(h): Hes1 mRNA in the RDME model with voxel volume $50 \mu m^3$, i.e., less levels of noise. (i): the average Hes1 mRNA at low volume, $1 \mu m^3$, (dashed line) and high volume, $50 \mu m^3$, (solid line) over time where the horizontal lines denote the times at which the spatial dynamics are shown in the top and middle rows. Blue and orange, again, denote cells with high and low expression and boundary and initial conditions are chosen as previously only this time initial conditions are in number of molecules. Based on a mouse embryonal stem cell volume of approximately $50 \mu m^3$ (size based on Pillarisetti2009Wang2011 assuming spherical cells) and a mean number of $8104$ molecules per cell Ho2018abundance, we find our results in $\mu M$.
• Figure 3.1: Fix point arguments. Left: the unique homogeneous stationary state is the fix point $\bar{x}_0 = \varphi(\bar{x}_0)$. Right: if $\gamma_2'(\bar{x}_0) > 1$, then there are cyclic (non-homogeneous) solutions $\bar{x}_1 < \bar{x}_0 < \bar{x}_2$.

Theorems & Definitions (16)

• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Lemma 3.5
• Proposition 3.6
• proof