The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster
Introduction
Understanding how genetic information is translated into proteins in the correct temporal sequence and spatial location to produce the various cell types in an adult organism remains a major challenge in developmental biology. The sequencing of the genome of various organisms provides the first step toward this understanding, but it is equally important to understand the patterns of control involved in gene regulation. Gene products often regulate their own production or that of other proteins at any of a number of steps, and frequently the result is a complex network of regulatory interactions. Recent experimental progress in dissecting the qualitative structure of many signal transduction and gene control networks (see, for example, Davidson et al., 2002) has produced a surge of interest in the quantitative description of gene regulation. Broadly speaking, the modeling approaches can be divided into two main groups. In the “discrete-state” approach each network node (mRNA or protein) is assumed to have a small number of discrete states and the regulatory interactions between nodes are described by logical functions similar to those used in programming. Typically time is also quantized, and the network model that describes how gene products interact to determine the state at the next time gives rise to a discrete dynamical system (Bodnar, 1997; Mendoza et al., 1999; Bodnar and Bradley, 2001; Sánchez and Thieffry, 2001; Yuh et al., 2001). A more detailed level of description is used in the “continuous-state” approach, in which the level of mRNAs, proteins, and other components are assumed to be continuous functions of time, and the evolution of these components within a cell is modeled by differential equations with mass-action kinetics or other rate laws for the production and decay of all components (Reinitz and Sharp, 1995; von Dassow et al., 2000; Gursky et al., 2001). While it is widely thought that merely specifying the topology of a control network (i.e. the connections between nodes) places few constraints on the dynamics of the network, our purpose here is to demonstrate that in one well-characterized system, knowledge of the interactions together with their signatures, by which we mean whether an interaction is activating or inhibiting, is enough to reproduce the main characteristics of the network dynamics.
The genes involved in embryonic pattern formation in the fruit fly Drosophila melanogaster, as well as the majority of the interactions between them, are known (for recent reviews see Ingham and McMahon, 2001; Sanson, 2001; Hatini and DiNardo, 2001). As in other arthropods, the body of the fruit fly is composed of segments, and determination of the adult cell types in these segments is controlled by about 40 genes organized in a hierarchical cascade comprising the gap genes, the pair-rule genes, and the segment polarity genes (Hooper and Scott, 1992). These genes are expressed in consecutive stages of embryonic development in a spatial pattern that is successively more precisely defined, the genes at one step initiating or modulating the expression of those involved in the next step of the cascade. While most of these genes act only transiently, the segment polarity genes are expressed throughout the life of the fly. The segment polarity genes refine and maintain their expression through the network of intra- and intercellular regulatory interactions shown in Fig. 1. The stable expression pattern of these genes (specifically the expression of wingless and engrailed) defines and maintains the borders between different parasegments (the embryonic counterparts of the segments) and contributes to subsequent developmental processes, including the formation of denticle patterns and of appendage primordia (Hooper and Scott, 1992; Wolpert et al., 1998). Homologs of the segment polarity genes have been identified in vertebrates, including humans, which suggests strong evolutionary conservation of these genes.
The segment polarity genes encode for the transcription factor engrailed (EN), the cytosolic protein cubitus interruptus (CI), the secreted proteins wingless (WG) and hedgehog (HH), and the transmembrane receptor proteins patched (PTC) and smoothened (SMO) involved in transduction of the HH signal.
The pair-rule gene sloppy paired (slp) is activated before the segment polarity genes and expressed constitutively thereafter (Grossniklaus et al., 1992; Cadigan et al., 1994). slp encodes two forkhead domain transcription factors with similar functions that activate wg transcription and repress en transcription, and since they are co-expressed we designate them both SLP. The wg gene encodes a glycoprotein that is secreted from the cells that synthesize it (Hooper and Scott, 1992; Pfeiffer and Vincent, 1999), and can bind to the Frizzled (FZ) receptor on neighboring cells. Binding of WG to the FZ receptors on adjacent cells initiates a signaling cascade leading to the transcription of engrailed (en) (Cadigan and Nusse, 1997). EN, the homeodomain-containing product of the en gene, promotes the transcription of the hedgehog gene (hh) (Tabata et al., 1992). In addition to the homeodomain, EN contains a separate repression domain (Han and Manley, 1993) that affects the transcription of ci (Eaton and Kornberg, 1990) and possibly ptc (Hidalgo and Ingham, 1990; Taylor et al., 1993). The hedgehog protein (HH) is tethered to the cell membrane by a cholesterol linkage that is severed by the dispatched protein (Ingham, 2000), freeing it to bind to the HH receptor PTC on a neighboring cell (Ingham and McMahon, 2001). The intracellular domain of PTC forms a complex with smoothened (SMO) (van den Heuvel and Ingham, 1996) in which SMO is inactivated by a post-translational conformational change (Ingham, 1998). Binding of HH to PTC removes the inhibition of SMO, and activates a pathway that results in the modification of CI (Ingham, 1998). CI contains at least three distinct domains: an NH2 terminal region characteristic of transcriptional repressors, a zinc finger domain, and a COOH region typical of activation domains in transcription factors (Alexandre et al., 1996). The CI protein can be converted into one of two transcription factors, depending on the activity of SMO. Several proteins have been implicated in this conversion, including Fused, Suppressor of Fused, Costal-2, Protein kinase A, Slimb and the CREB-binding protein (Aza-Blanc and Kornberg, 1999; Lefers et al., 2001). When SMO is inactive, CI is cleaved to form CIR, a transcriptional repressor that represses (Aza-Blanc and Kornberg, 1999) and hh transcription (Ohlmeyer and Kalderon, 1998; Méthot and Basler, 1999). When SMO is active, CI is converted to a transcriptional activator CIA that promotes the transcription of wg and ptc (Alexandre et al., 1996; von Ohlen and Hooper, 1997; Méthot and Basler, 1999; Aza-Blanc and Kornberg, 1999).
The gene networks governing embryonic segmentation in Drosophila have been modeled using either the discrete-state approach (Bodnar, 1997; Bodnar and Bradley, 2001; Sánchez and Thieffry, 2001), or the continuous-state approach (Reinitz and Sharp, 1995; Gursky et al., 2001), but the first modeling work focusing on the segment polarity gene network was done by von Dassow and collaborators (von Dassow et al., 2000; von Dassow and Odell, 2002). von Dassow et al., developed a continuous-state model of the core network of five genes ( and hh) and their proteins. The initial choice of network topology failed to reproduce the observed expression patterns for these genes, but the introduction of two additional interactions led to surprisingly robust patterning with respect to variations in the kinetic constants in the rate laws. The robustness of this model with respect to the changes in the reaction parameters suggests that the essential features involved are the topology of the segment polarity network and the signatures of the interactions in the network (i.e. whether they are activating or inhibiting), and our first objective is to test this prediction.
The Boolean model we develop here is not based on continuous concentrations of mRNAs and proteins; only two states are admitted for each component corresponding to whether or not they are present. This choice is motivated by the ON/OFF character of the experimentally observed gene expression patterns.1 We show that this model reproduces the wild-type gene expression pattern, as well as the ectopic patterns corresponding to various mutants. We determine all the steady states of the model analytically, and we find that there is a surprisingly small number (6) of distinct ones, half of which are observed experimentally. We find the domains of attraction of these steady states by a systematic search in the space of possible initial conditions. Furthermore, we determine the minimal pre-patterning necessary to produce the wild-type spatial expression pattern, and thereby show that the majority of non-initiation errors in the pre-pattern can be corrected during the temporal evolution from such states. The model demonstrates the remarkable robustness of the segment polarity network, and shows that the robustness resides in the topology of the network and the signature of the interactions.
Section snippets
Description of the Boolean model
In the model, each mRNA or protein is represented by a node of a network, and the interactions between them are encoded as directed edges (see Fig. 1).2 The state of each node is 1 or 0, according as the corresponding substance is present or not. The states of the nodes can change in time, and the next state of node i is
Functional topology of the segment polarity network
The rules for advancing the current state of the network given in Table 1 can be used to construct an expanded graph that reflects the function of the network. Consider the transcription of the hh gene. Fig. 1 shows that hh has two incoming edges, one from EN and one from CIR, and Table 1 shows that transcription of the hh gene requires both the presence of the EN protein and the absence of the CIR protein. To represent this conjunction in a graphical form, one can say that hht+1 depends on the
Comparison between numerical and experimental results
The segment polarity genes are activated by the pair-rule genes in the cellular blastoderm phase (stage 5 according to the classification of Campos-Ortega and Hartenstein, 1985) of Drosophila embryogenesis, and maintain the parasegment borders and later the polarity of the segments from the end of gastrulation through germ-band elongation (stages 8–11, see Wolpert et al., 1998). The parasegment borders form between the wg and en/hh expressing cells (Hooper and Scott, 1992; Wolpert et al., 1998
Determination of the steady states and their domains of attraction
The fact that the Boolean model reproduces the results of numerous experiments remarkably well suggests that the structure of the model is essentially correct, and warrants exploration of problems that have not been studied experimentally. For example, we can determine the complete set of stable steady-state patterns of segment polarity gene expression, and estimate the domain of attraction of these states. The former can be done analytically by noting that these are fixed points of the
A two-step model
While the steady states of the model reproduce experimentally-observed expression patterns, the temporal evolution may not reflect the in vivo evolution of expression patterning. We assume that the expression of mRNAs/proteins decays in one time step if their transcriptional activators/mRNAs are switched off, and this assumption can induce transient on-off flickering in the expression pattern. Such flickering can be eliminated by relaxing the assumptions of immediate switch-off, and assume more
Expression of the segment polarity genes after a round of cell division
When defining the Boolean functions characterizing the interactions between nodes, we have assumed that the transport of WG and HH can be disregarded. One of the reasons for doing so is the fact that we assume that the parasegments are four cells wide, as they are at the beginning of germ band elongation. Our results indicate that if we allow the spreading of WG and HH further than the membrane of the nearest neighbors of the cells producing them, ectopically broad wg, en, and hh stripes
Discussion and outlook
Since this work focuses on the regulatory network of the same genes as in von Dassow et al. (2000), it is worthwhile to discuss the common points and differences between our results. Our model, based on the topology and signature of the interactions between the segment polarity genes, confirms their suggestion that the topology of the network has a dominant role in its function. But it should be noted that the topology used in the von Dassow et al. (2000) model and the present work is
Acknowledgements
This work was supported in part by NIH Grant #GM 29123.
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