Fig 1.
Mathematical modeling of nuclear Dorsal gradient in Drosophila embryos.
(A) Schematic representation of the Reaction-Diffusion Network Model regulating nuclear Dorsal (nDl) localization. Kinetic constants k1 and k2 mediate synthesis and Toll-independent degradation of the inhibitor Cactus, respectively. Dl dimers can enter and leave the nucleus by direct flow (k3 and k4), independent of Toll activation. k5 and k6 mediate reversible binding between cytoplasmic Dl dimers (cDl0) and free Cactus (Cf) to form trimeric complexes (DlC). DlCT complex is reversibly formed by the interaction between DlC and activated Toll (T) membrane receptor (k7 and k8). Following DlCT complex formation, Toll activation induces Dl and Cactus phosphorylation, releasing Dl and C from the complex. This irreversible reaction is controlled by k9. Cytoplasmic phosphorylated Dl dimers (cDl*) enter the nucleus (nDl*) while phosphorylated and ubiquitinated Cactus (Cub) is degraded by the proteasome (k10). k12 controls nDl* output from the nucleus. Note that T represents only the activated form of the Toll receptor. (B) Detailed reaction network stoichiometry. (C) Key relationships among model species. Total Cactus (C) is the sum of all species that contain Cactus. Total nuclear Dorsal (nDl) is the sum of nDl0 and nDl*. Total cytoplasmic Dorsal (cDl) includes free cDl, cDl*, plus the two DlC and DlCT complexes. Total Dorsal is the sum of nDl and cDl. The model was solved for cleavage cycle 14 embryos.
Fig 2.
The Reaction-Diffusion Network Model reproduces the wild type nDl gradient profile and discriminates two different Dl nuclear entry modes.
(A) Optical section of a wild type (WT) embryo stained for Dorsal protein. (B) Nuclear Dl fluorescence intensity was extracted from sections as in A, measured and plotted as half gradients (circle). The black curve displays model simulation. The y-axis represents nDl fluorescence intensity along the ventral-to-dorsal (V-D) embryonic axis (x-axis). Data are mean ± s.e.m. (C) High to low nDl levels (red circles) define different DV territories: ventral mesoderm represented by snail expression (green), lateral neuroectoderm represented by short gastrulation (sog, magenta) and dorsal ectoderm defined by decapentaplegic expression (dpp, grey). (D) Simulations discriminate nuclear Dl that enters the nucleus by direct flow (nDl0, dotted curve) or induced by Toll (nDl*, dashed curve). The black curve indicates total nDl model simulation, as in B. Ventral (V) region to the left, dorsal (D) to the right.
Table 1.
Dimensionless model parameters.
Fig 3.
Dl gradient simulations are responsive to variations in Dl levels.
(A) Simulation (solid curves) and experimental data (circle symbols) from mutant (orange) and wild type (black) embryos were plotted in the same graph. Nuclear Dl gradient from dl6/+ mutant embryos is simulated and fitted using an 85% reduction in total Dorsal protein level. (B) Spatial distribution of Dorsal protein that enters the nucleus by direct flow (nDl0, dotted curve), or Toll induced (nDl*, dashed curve) and total nDl. (C) Distribution of DlC and DlCT species amounts for wild type (black) and dl6/+ (orange) genotypes. (D) nDl gradient simulations resulting from 20% (yellow), 40% (green), 60% (blue), 80% (brown) Dorsal protein reductions compared to a wild type nDl gradient (black).
Table 2.
Dimensionless parameters for each mutant.
Fig 4.
Cact produces distinct effects along the DV axis by controlling unbound versus complexed Dl elements.
(A) Simulations (solid curve) and experimental data (circle symbols) from cactus mutant (blue) and wild type (black) embryos. Nuclear Dl gradient from cactA2/cact011 mutant embryos are simulated and fitted using a 55% reduction in Cact protein. (B) Spatial distribution of Dorsal that enters the nucleus by direct flow (nDl0, dotted curve), by Toll induced (nDl*, dashed curve) and total nDl. (C) Distribution of Total cDl (defined as cDl0 + DlC + DlCT + cDl*; star symbols), sum of the unbound cDl species (cDl0 + cDl*) and the DlC + DlCT complexes for wild type (black) and cactA2/cact011 (blue) genotypes. (D) nDl gradient simulations resulting from 100% (magenta, constant nDl), 20% (yellow), 40% (green), 60% (blue), 80% (brown) Cactus protein reductions are compared to wild type nDl gradient (black).
Fig 5.
A dl/cact double mutant highlights the positive role of Cact on Dl nuclear translocation.
(A) Simulations (solid curves) and experimental data (circle symbols) from mutant (purple) and wild type (black) embryos. Nuclear Dl gradients from dl6/cactA2 mutant embryos were simulated using simultaneous reduction of 50% Cactus and 70% Dorsal. (B) Spatial distribution of Dorsal that enters the nucleus by direct flow (nDl0, dotted curve), by Toll induced (nDl*, dashed curve) and total nDl are shown in wild type (black) and mutant (purple) simulations. (C) Distribution of Total cDl (defined as cDl0 + DlC + DlCT + cDl*; star symbols), sum of the unbound cDl species (cDl0 + cDl*) and the DlC + DlCT complexes for wild type (black) and dl6/cactA2 (purple) genotypes.
Fig 6.
Variations in nDl gradient slope disrupts the precision of target gene expression DV territories.
(A) Cross-sections of wild type (WT) and mutant (dl6/cactA2) cleavage cycle 14 embryos hybridized with snail (green) and sog (magenta) antisense RNA probes. Nuclei were stained using DAPI (blue). Note sog transcripts invading the ventral territory in dl6/cactA2 mutants (zoomed images). (B) Derivatives of nDl concentration (y-axis) as a function of the position along the embryonic DV axis (x-axis) for WT (black), dl6/+ (orange), dl6/cactA2 (purple), and cactA2/cact011 (blue) mutant embryos. (C) Peak, basal levels, amplitude and highest slope of nDl distribution for each different genetic background.
Fig 7.
Model simulations reproduce Toll-dependent and Toll independent pathway mutant conditions.
(A-C) Simulations for simultaneous reduction of kinetic constants k9 and k10 (0%, 10% and 50% reductions), showing the distribution of nuclear Dorsal (A), DlCT complex (B) and DlC complex (C). (D-F) Simulations decreasing activated Toll receptor (Tlact) to 10% or 50% or inactivated Toll (Tlin), showing the distribution of nuclear Dorsal (D), DlCT complex (E) and DlC complex (F). (G-L) Simulations increasing kinetic constant k9 (G-I) or k5 (J-L), by 3x, 2x or 1.5x, in a cactA2/cact011 mutant background. Shown are nDl (total nuclear Dorsal dimers, G, J), nDl0 (free nuclear Dorsal dimers; H, K) and nDl* (nuclear Dorsal dimers activated by Toll Pathway; I, L). G and J also display experimental data from the cactA2/cact011 mutant (circles) for comparison with wild type and mutant simulations. All concentrations are plotted along the V-D axis.