Fig 1.
Biological models for cell signalling in the intestinal crypt epithelium.
Simplified schematics depict: (A) the Notch pathway, where labels S2-S4 indicate cleavage sites in the Notch receptor; (B) a cross-section of a crypt in the large intestine. Arrows denote the general direction of cell migration. Cells are coloured according to their type: red indicates a stem cell; purple indicates a proliferative progenitor or transit cell; secretory goblet cells are shown in green, and absorptive enterocytes are shown in blue; (C) the Wnt pathway in its low-stimulus state; (D) the Wnt pathway in its high-stimulus state. All network diagrams depict the cytoplasm in green, the nucleus in blue, the intercellular space in grey and the proteasome in red, while transcriptional targets of the pathway are listed in red. Image (B) is adapted from Reizel et al. [8], originally published by PLOS and provided under a Creative Commons Attribution Licence, CC-BY-2.5.
Fig 2.
Network representation for our model of Notch-Wnt interaction.
Our model comprises: (A) the Wnt pathway, (B) the Notch pathway, and (C) crosstalk points. Numbered steps are justified in S1 Text. Major steps involving the β-catenin crosstalk hub are shown in red, while Wnt-dependent steps are shown in purple. Black AND gates signify the formation of intermediate complexes from two molecular partners. Circular end caps indicate inhibition steps.
Table 1.
Chemical reaction network results for decoupled and full system.
Concordance results for decoupled and whole networks in a homogeneous system, analysed using the CRN Toolbox [100]. A tick indicates a concordant network (monostable) and a cross, a discordant network (multistable). The second column indicates which β-catenin crosstalk points were included in each network. β-catenin is not consumed in Step 14 and so the comparison of the two coupling points does not apply to the Wnt-only system.
Fig 3.
Plots showing how variation in θ2, the proportion of Notch-mediated control of the Hes1 promoter, affects the system dynamics.
(A) Cross-section schematic of a crypt from the large intestine; image adapted from Reizel et al. [8], originally published by PLoS and provided under a Creative Commons Attribution Licence, CC-BY-2.5. (B–E) Influence of θ2 upon the Hes1 steady state at W = 0.0 (black), W = 1.0 (red) and W = 2.0 (blue); θ2 ∈ [0, 1] represents the proportion of Notch-mediated transcription of Hes1. Timecourses show Hes1 and β-catenin expression for healthy cell pairs, for (B) θ2 = 0.00 (i.e. entirely Wnt-mediated), (C) θ2 = 0.55, (D) θ2 = 0.75, (E) θ2 = 1.00 (i.e. entirely Notch-mediated). The timecourse for the first cell of each pair is indicated by a solid line; the second, by a dotted line. Where only one timecourse is apparent, the cell pair is synchronised. Standard initial conditions and parameters are used, stated in Tables S.3–S.7 of S1 Text; the ODE model comprises equations (S.1)–(S.13) of S1 Text.
Fig 4.
Dynamic timecourses showing cell pair responses to mutation in one cell.
(A) Cross-section schematic of a crypt from the large intestine; image adapted from Reizel et al. [8], originally published by PLoS and provided under a Creative Commons Attribution Licence, CC-BY-2.5. (B–E) Timecourses for cell pairs started from homogeneous initial conditions; all cells are healthy at start of simulation. (B) Healthy cell pair; (C) APC mutants (dashed line) acquire their first APC knockout (ρAPC = 0.5) at t = 12h and the second (ρAPC = 0.0) at t = 24h; (D) hyperstimulated Wnt mutants (dashed line) transform to a W = 2.0 state at t = 12h. Except for Wnt mutants, all plots in panels (B)–(D) indicate simulations with (red) W = 1.0 and (black) W = 0.0. Inset panel (E) compares the Hes1 expression in the healthy and APC mutant scenarios for W = 0, indicated by the asterisks. In this case, the healthy scenario is shown in black and the APC mutant in red. Standard initial conditions and parameters are used, stated in Tables S.3–S.7 of S1 Text; the ODE model comprises equations (S.1)–(S.13) of S1 Text, with APC modifications to Eqns. (S.9) and (S.10) of S1 Text for study (C).
Fig 5.
Proof-of-concept simulations for stochastic evolution of Hes1.
The black timecourse indicates the outcome from the deterministic model; red lines indicate the results of ten separate stochastic runs, with mean given by the blue line. The addition of extrinsic noise to the Hes1 production term changes the outcome from a lower to a higher expression level of Hes1 in three out of the ten cases.
Fig 6.
Notch parametrisation outcomes.
(A) Response of oscillation period in cell 1 to variations in initial conditions in a two-cell system running the decoupled, dimensional Notch model, Eqns.(S.1)–(S.6) of S1 Text. Statements of initial conditions of the form [x, y] indicate that all seven Notch species in cells 1 and 2 are initialised to x and y respectively. Owing to symmetry considerations, only the oscillations in Cell 1 were measured; results for Cell 2 correspond to a reflection of this surface in the line y = x. An amplitude filter was applied during generation of the plot, to disregard any small-amplitude oscillations (< 0.001) arising from the computational solution process, rather than true oscillations of the ODE model. (B) Diagonal entries of (A) yield homogeneous evolution with damped oscillations, as in this timecourse of a cell pair from initial conditions (0.5, 0.5). (C) Off-diagonal entries of (A) show heterogeneous evolution, as in this timecourse of a cell pair from initial conditions (0.1, 0.5).
Fig 7.
Timecourse for β-catenin evolution in the dimensional, decoupled Wnt submodel, Eqns. (S.9)–(S.13) of S1 Text (line graph), compared against the experimental readings of Hernández et al. [104] (point data). The timecourse for the Wnt submodel uses the parameter values listed in Tables S.3–S.6 of S1 Text, and the initial conditions of Table S.7 of S1 Text.