Conceived and designed the experiments: DU JH AR MG CF. Performed the experiments: DU JH. Analyzed the data: DU JH AR MG. Contributed reagents/materials/analysis tools: CF. Wrote the paper: DU JH AR MG CF.
The authors have declared that no competing interests exist.
The axial bodyplan of
Embryonic development in
The gap gene system is one of the most widely studied morphogen systems in
The model representation of the gap gene network. The network topology in (
Most inferences regarding the gap genetic regulatory network (GRN) have been drawn from mutant and gene dosage studies in which the effects on morphology, gap, pairrule, or segment polarity genes are observed
Previous 1D PDE models have been used effectively to infer network topology and investigate patterning regulation
Quantitative spatiotemporal atlases of gene expression data in the
Using the VE data, we evaluate the impact of 1D model assumptions, conversion from 1D to 3D geometries, and incorporation of fully 3D protein distribution data in model simulation. Herein we reconstruct the 1D gap gene model of Jaeger
Model  Geometry  Initial Conditions  Optimal GRN Parameters 

1D domain representing partial 35%–92% AP axis  Gt_{0} = Kni_{0} = Kr_{0} = Tll_{0} = 0; Bcd_{SS}, Hb_{0}, Cad_{0} values in Jaeger 


1D domain representing full 0%–100% AP axis  Gt_{0} = Kni_{0} = Kr_{0} = Tll_{0} = 0; Bcd_{SS}, Hb_{0}, Cad_{0} values in Jaeger 


VE 3D domain  Gt_{0} = Kni_{0} = Kr_{0} = Tll_{0} = 0; Bcd_{SS}, Hb_{0}, Cad_{0} values in Jaeger 


VE 3D domain  Gt_{0} = Kni_{0} = Kr_{0} = Tll_{0} = 0 Hb_{0,} Cad_{0} values in Jaeger 


VE 3D domain  Gt_{0} = Kni_{0} = Kr_{0} = Tll_{0} = 0; Hb_{0,} Cad_{0} values in Jaeger 

In addition to GRN sensitivities highlighted by previous 1D analyses
Another question addressed in this study is whether inclusion of 3D data improves optimization by the inclusion of additional constraints without increasing the degrees of freedom in the model. Unexpectedly, we found that the incorporation of additional 3D information in the form of a realistic DVasymmetric Bcd worsened the error between optimized 3D models and data. This suggests the involvement of additional regulators in the formation of DVasymmetries and indicates a direction for future modeling studies.
Before analyzing the effects of embryonic geometry and DVasymmetric positional information, we reimplemented the 1D model of Jaeger
Model output was simulated over a 0–100% AP length domain using the optimal GRN reported by Jaeger
Though
Pronounced shifts in Tll and Kr distributions, coupled with the qualitative change in the anterior Gt distribution, demonstrate the role boundary conditions play in the in the distribution of gap gen products for a given set of parameters. Though the output of
Though domain boundary placement affects the banding pattern,
To facilitate a direct comparison between 1D and 3D models presented herein, we first evaluated the goodnessoffit between the 35–92% AP (
Though
Beginning with the GRN optimized on the full 1D domain, we extended the model to a 3D domain using the geometry in the VE. This was performed by implementing the system of PDEs on a 2D surface “wrapped” around the VE geometry. We used this model to evaluate the effects of both model geometry and DVasymmetric initial conditions on model output.
To assess the effects of model geometry on patterning independent of initial conditions, the model was first simulated using DVsymmetric initial conditions (
Initial conditions in various models. (
Simulation results in the 3D model. (
Though there are some DVasymmetries present in the output (e.g., slight curvature of the anterior Gt stripe), 1D versus 3D domain geometry alone has only a modest impact on DV patterning of gap genes. This suggests that the pronounced DVasymmetries present in the final distributions of the proteins at the onset of nuclear division 14 (
To evaluate the impact of DVasymmetric inputs on the model, we modified the steadystate Bcd distribution shown in
Evaluated at the optimal 1D GRN
The celltocell variability in patterning found for many simulated proteins (e.g., Gt, Cad, and Kni) in
Having observed that a GRN inferred on the 1D domain (and lacking DV asymmetries) produces a qualitatively incorrect fit compared to 3D data, we attempted to optimize the GRN with Matlab's constrained search function
In the 3D regime,
Parametric noise alters model output. Lateral view of VE geometry for all genes is shown in rows
>In summary, the GRNs we inferred in this study are qualitatively similar: magnitudes of parameters vary by approximately 10% and parameter sign stays the same in all but a few lowmagnitude parameters (see
The understanding of
Before comparing 1D and 3D geometries, we examined the effect of boundary position in PDE solutions. Though embryos do not contain physical barriers to diffusion at 35% and 92% of the AP axis, small spatial gradients (
Prior analyses demonstrated the sensitivity of gap gene models to GRN parameter values
In addition to the parameter sensitivity and boundary conditions, our work also demonstrate the use of accurate 3D geometry and its effects on model predictions. We found that geometry alone has a limited effect on gap gene patterning: Excepting slight DVasymmetry brought about by the curvature of the 3D embryo, 1D output from
Though the 3D embryonic geometry was insufficient to explain DVasymmetries in gap gene data, it allowed us to explore the effect of DVasymmetric protein distributions on patterning. Notably, the 1D Bcd distribution of
When considering 3D models and the data associated with them, we endeavored to identify any constraints on model optimization. This model has many degrees of freedom and additional information encoded in the DV asymmetries of gap genes might better guide parameter searches toward accurate GRNs. However, we observed no improvement in RMSE values and failed to find any novel GRNs for DVasymmetric models.
Though our ensemble of models has led to interesting findings, we acknowledge model limitations. Recent modeling studies recognize that Cad and Tll patterning cannot be completely accounted for by gap genes in existing models; maternal mRNA complicates Cad expression and Tll is under the regulation of additional proteins
The primary focus of this work is the comparison of 1D and 3D model geometries.
Finally, two cases of DV model mismatch suggest modifications that could be incorporated into future models. First, anterior Gt is more highly expressed on the dorsal side of the embryo
Building on the successful 1D/3D embryonic modeling approach of Umulis
Note that previous 1D models were simulated by the spatiallydiscretized ordinary differential equations using the finite difference method: concentrations were tracked at uniformlydistributed nodes (nuclei) along the AP axis and diffusive fluxes across the Δ
We developed two FEM meshes on which to simulate spatiotemporal gap gene evolution. A 1D linear domain represents the 35–92% AP axis, and replicates the domain used in previous models
Though the 3D domain is a closed surface without AP flux boundaries, the partial (
Numerical integration of PDEs requires specification of initial conditions as well as boundary conditions. For purposes of model comparison, we chose initial conditions specified in previous models
While the Bcd data provided by Jaeger
Spatiotemporal regulation of gap gene expression spans the mitotic nuclear division between nuclear cycle 13 and 14a. For purposes of comparison, we chose to simulate the same timecourse as previous models. We begin by simulating the conclusion of cycle 13 for sixteen minutes, mitosis for five minutes, and continue to simulate cycle 14a for the remaining fortynine minutes
This set of initial and boundary conditions, coupled with the reactiondiffusion equations and a geometric domain, constitutes a numerically soluble model. To calculate model error, we used a straightforward root mean squared error cost function:
Using the cost function (eqn. 5), we optimized the full 1D and 3D models against scaled VE data using the Optimization Toolbox in MATLAB R2009a
We used the GA as implemented in MATLAB. The population of size twenty genomes (parameter sets) was initialized with nineteen randomized parameter sets and the locallyoptimized parameter set found for
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