Figure 1.
Gap gene genetic regulatory network.
The model representation of the gap gene network. The network topology in (A) represents negative (black box, flat line) and positive (white box, arrowhead line) regulatory effects on each target gene (blue). Dashed lines represent near-zero regulatory inputs that may be negligible. This qualitative topology is quantified in (B) as a set of genetic regulatory network (GRN) weight parameters wb,a, the influence of gene b on gene a. From left to right, each set of seven inputs represent Cad, Gt, Hb, Kni, Kr, Tll, and Bcd. Each cluster of seven interactions represents a target gene Cad, Gt, Hb, Kni, Kr, and Tll.
Table 1.
Model Variants and Corresponding Optimal Parameter Sets.
Figure 2.
One-dimensional model results.
Model output was simulated over a 0–100% AP length domain using the optimal GRN reported by Jaeger et al. Solid vertical lines represent the original model boundaries, not used in this simulation. (A) (solid lines) shows qualitative agreement with the Jaeger model
(dashed lines) in the 35–92% AP range, but shows discrepancies at either end of the domain due to the movement of boundaries; all species displayed at t = 70 min. (B) The best-fit GRN from Jaeger et al. was locally optimized to improve the agreement of the 0–100% AP length, model
(solid lines), and the original Jaeger et al. original model (
dashed lines); all species displayed at t = 70 min. (C) VE protein data for Gt, Hb, Kni, Kr at t = 70 min; VE mRNA data for Tll at t = 70 min; protein data from Jaeger et al. for Cad at t = 56 min. (D) Model output (
) was also optimized against VE data (RMSE = 13.992); Gt, Hb, Kni, Kr, Tll at t = 70 min; Cad at t = 56 min. Despite modest improvements in model agreement in the 35% and 92% region (C–D), the resulting changes in parameter values were small. (E) Optimized parameter magnitudes vary but signs remained the same in most cases (blue -
; green -
; red -
).
Figure 3.
Initial conditions in various models. (A) 1D model initial conditions, reported by Jaeger et al., and used in models and
. (B) 1D initial conditions were mapped onto the 3D embryonic geometry (
). (C), 1D initial Cad protein distribution, (D) 1D initial Hb protein distribution. Subsequent models incorporated (E) DV-asymmetric interpolated [Bcd] distribution (
) or (F) smoothed DV-asymmetric interpolated [Bcd] distribution (
).
Figure 4.
Three-dimensional model results.
Simulation results in the 3D model. (A–H) Lateral view of VE geometry is shown in rows A–G (Gt, Hb, Kni, Kr, Tll at t = 70 min, Cad at t = 56 min); row H displays RMSE difference between model and VE data summed with all time points. Column 1 shows scaled VE data. Column 2 displays output from evaluated with GRN
. Column 3 contains output from
incorporating DV-asymmetric Bcd data and GRN
; Column 4 illustrates the effect of the smoothed Bcd interpolant in
while considering the same GRN
. Column 5 displays output from
with reoptimized parameters
. White boxes indicate the lateral areas where Jaeger et al. optimized their 1D model. Animations of pattern development are available for column 2 (
, Movies S1, S2, S3, S4, S5, S6) and column 5 (
, Movies S7, S8, S9, S10, S11, S12) in the supplementary material.
Figure 5.
Model is not robust to noise in GRN parameters.
Parametric noise alters model output. Lateral view of VE geometry for all genes is shown in rows A–G (all outputs at t = 70 min). Each column displays output at t = 70 min evaluated with GRN
. Columns 2–5 represent randomly chosen sample output when a normally distributed noise vector ε is added to the GRN parameter set (denoted θ). ε has mean of 0 and variance that scales with θ.