Robustness under functional constraint: the genetic network for temporal expression in Drosophila neurogenesis

Abstract

Precise temporal coordination of gene expression is crucial for many developmental processes. One central question in developmental biology is how such coordinated expression patterns are robustly controlled. During embryonic development of the Drosophila central nervous system, neural stem cells called neuroblasts express a group of genes in a definite order, which leads to the diversity of cell types. We produced all possible regulatory networks of these genes and examined their expression dynamics numerically. From the analysis, we identified requisite regulations and predicted an unknown factor to reproduce known expression profiles caused by loss-of-function or overexpression of the genes in vivo, as well as in the wild type. Following this, we evaluated the stability of the actual Drosophila network for sequential expression. This network shows the highest robustness against parameter variations and gene expression fluctuations among the possible networks that reproduce the expression profiles. We propose a regulatory module composed of three types of regulations that is responsible for precise sequential expression. This study suggests that the Drosophila network for sequential expression has evolved to generate the robust temporal expression for neuronal specification.

Figure 1. Sequential expression of temporal transcription factors within neuroblasts in the Drosophila CNS.

(A) The relative position of neuroblasts (NBs) in Drosophila embryo. The picture is the ventral view of NBs and shows Cas expression in the NBs at developmental stage 12. The bracket indicates a single segment. Dashed line corresponds to the midline. Scale bar: . (B) The expression levels of Hb, Kr, Pdm, and Cas in a single NB (NB 2–4 lineage) are shown from the developmental stage 10 to 12: early stage 10 (st. 10), early stage 11 (e11), mid stage 11 (11), late stage 11 (l11), mid stage 12 (12), late stage 12 (l12). (C) Schematic representation of the change of the expression pattern in a single NB. (D) The expression profiles of WT, loss-of-function, and overexpression mutants of the genes observed in the experiments (for references, see Table 2). (E) Reconstructed genetic network for sequential expression in Drosophila NBs. Repression from hb to cas (dashed line) was suggested to exist , although there is no direct verification. When the Drosophila network is invoked in this article, this regulation is also included. (F) Matrix representation of the Drosophila network.

Figure 2. Reconstructed Drosophila network cannot reproduce the experimentally reported expression profiles.

Sequential gene expression of reconstructed Drosophila network is simulated using Boolean model. The grids filled with colors represent ON states of the genes. The dynamics could be different depending on the choice of the default expression states . (A) ; (B) ; and (C) .

Figure 3. Architecture of the detected functional networks.

(A) Architecture of the functional networks reproducing the gene expression profiles observed in the experiments. The black arrows are the regulations that appear in all the functional networks. The brown arrows are the regulations from the presumptive factor x that also appear in all the functional networks. The other regulations existing in the actual Drosophila network are shown by gray arrows. (B) Matrix representation of the functional networks. Elements of {J_ij} are shown as either + for activation, − for repression, or 0 for the absence of regulation. (C) Frequency distributions of the distances of networks from the Drosophila network. The distributions are drawn from the functional networks (N = 384; magenta), all the possible networks (N = 14,348,907; blue), and the networks randomly reconnected from the functional ones (N = 38,400; yellow). From each of the functional networks, 100 reconnected networks were generated. The regulatory interactions from x and positive self-feedbacks are neglected in counting the number of different regulations.

Figure 4. Temporal dynamics of the Drosophila network in the continuous model.

The dynamics of expression levels of proteins {P_i(t)} with different parameter values (upper) and discretized representation of a typical temporal dynamics (lower). In addition to the known genes, the presumptive factor x is also incorporated. The expression level of X changes from a high level to a low level as in the previous model. Each gene is considered to be in the ON state when the expression level is larger than a threshold P_th. The parameter values of and {S_i} are randomly selected from the following ranges: for and for ; and and . The other parameter values are set as shown in Table 4.

Figure 5. Robustness of the gene expression profiles in the functional networks.

(A) The fraction of trials that reproduce the experimental expression profile against random assignments of parameters. The values of , , and are randomly chosen within the ranges shown in Table 4. The other fixed parameter values are also listed in Table 4. Neglecting the positive self-feedback regulations in the 384 functional networks, 120 networks were chosen and investigated ( Materials and Methods ). The dynamics were checked for 50,000 trials in each network. The networks were sorted based on the distance from the Drosophila network (N_d). Here N_d corresponds to the number of regulations different from the Drosophila network. Because there are a few possible regulations from the unknown factor x, more than one network with N_d = 0 exist. (B) The fractions of the trials that reproduce the experimental profile under expression noise (vertical axis) are plotted against the fraction of successes against the random parameter assignments. To analyze the stability against noise, we used 1000 different parameter sets, by which the expression profile is reproduced in the absence of noise for each network. The dynamics were checked for 50 trials for each parameter set.

Figure 6. Contribution of the actual regulations to the robustness of the system.

(A) The fraction of the trials that reproduce the experimental WT expression against parameter variations. The data of Figure 5A are replotted for the Drosophila network, the networks lacking an indicated regulation (one of the gray arrows in Fig. 3A) and the minimum network (black and brown arrows in Fig. 3A). (B) The fractions of the trials that reproduce the experimental profile under gene expression noise with various intensities. We used 5,000 different parameter sets with which the profile is reproduced in the absence of noise. The dynamics are checked for 50 trials for each parameter set.

Figure 7. Graphical representation of parameter sets with which the WT sequential expression profile is reproduced.

(A) The Drosophila network, the networks lacking (B) activation from Kr to pdm, (C) activation from pdm to cas, (D) repression from hb to cas, and (E) the minimum network. The parameters involved in minimum network are shown. Each spoke represents a value of indicated parameter between the range used for random parameter assignment (Table 4). The value of is shown by normal scale and those of the other parameters are shown by log scale. Each polygon indicates one parameter set. Solid and broken lines indicate mean and s.d. of obtained parameters. The data are drawn from 5,000 trials of the random assignment of parameter values.

Figure 8. Parameter dependencies of robustness for the Drosophila network.

The fractions of successes for random assignment of parameter values are plotted under the different strengths of regulations (, , and ) and default promoter activities (S_pdm and S_cas). Dependencies of robustness to (A) (strength of activation from Kr to pdm) and S_pdm, (B) (strength of activation from pdm to cas) and S_cas, and (C) (strength of the repression from hb to cas) and S_cas. The other parameters are set as listed in Table 4. The temporal dynamics were tested for 50,000 trials.

Figure 9. Regulatory module for precise sequential expression.

The regulatory interactions and schematic expression profiles of the networks. (A) Regulatory interactions of the minimum network for sequential expression (left). This network reproduces the sequential expression under appropriate conditions (middle). However, the parameter variations from the appropriate values and the increase of noise could easily alter the expression profiles (right). (B) Regulatory interactions of the Drosophila network (left). Three types of regulations in this network enable the temporal expression in the precise order.