Abstract
All insects possess homologous segments, but segment specification differs radically among insect orders. In Drosophila, maternal morphogens control the patterned activation of gap genes, which encode transcriptional regulators that shape the patterned expression of pair-rule genes. This patterning cascade takes place before cellularization. Pair-rule gene products subsequently 'imprint' segment polarity genes with reiterated patterns, thus defining the primordial segments. This mechanism must be greatly modified in insect groups in which many segments emerge only after cellularization1. In beetles and parasitic wasps, for instance, pair-rule homologues are expressed in patterns consistent with roles during segmentation, but these patterns emerge within cellular fields2, 3, 4. In contrast, although in locusts pair-rule homologues may not control segmentation5, 6, some segment polarity genes and their interactions are conserved3, 7, 8, 9, 10. Perhaps segmentation is modular, with each module autonomously expressing a characteristic intrinsic behaviour in response to transient stimuli. If so, evolution could rearrange inputs to modules without changing their intrinsic behaviours. Here we suggest, using computer simulations, that the Drosophila segment polarity genes constitute such a module, and that this module is resistant to variations in the kinetic constants that govern its behaviour.
Gap and pair-rule gene products are nuclear proteins that form short-range gradients in the Drosophila syncytial blastoderm, locally modulating each other's expression through direct transcriptional control. In contrast, segment polarity genes refine and maintain their expression state through a network of cross-regulatory interactions that require cell–cell communication1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Many segment polarity genes, unlike gap and pair-rule genes, remain active throughout development; the segment polarity network remembers the pattern imprinted upon it, then provides positional read-outs for subsequent developmental processes, including specification of neuroblasts, denticle patterns and appendage primordia. Thus, the intrinsic behaviour of the putative segment polarity module consists of stable, reiterated, asymmetric expression of its constituent genes, especially of the principal outputs wingless (wg), hedgehog ( hh) and engrailed (en)15; in Drosophila the transient stimuli are pair-rule genes.
We used computer simulations to investigate whether the known interactions among segment polarity genes suffice to confer the properties expected of a developmental module. Given its probable conservation among diverse insects and perhaps beyond, we expected such a module to exhibit buffering against quantitative changes in gene function, and to be insensitive to the exact nature of input stimuli. We summarized (Box 1) the interactions among those segment polarity gene products that we reasoned might suffice to mimic the wild-type expression patterns of segment polarity genes (Fig. 1a). We abbreviated intermediate pathways and did not explicitly represent 'generic' components such as the transcriptional machinery. To formulate a dynamical model based on Box 1a required nearly 50 free parameters, including half-lives of messenger RNAs and proteins, binding rates, and cooperativity coefficients. The real values of these are unknown and the biologically realistic range for most parameters spans several orders of magnitude. Therefore we asked: is there any set of parameter values for which the network model exhibits the desired behaviour, given realistic initial conditions?
Figure 1: Segment polarity gene expression in Drosophila, pattern from wild type and several computer generated 'solutions'.
![Figure 1 : Segment polarity gene expression in Drosophila, pattern from
wild type and several computer generated |[lsquo]|solutions|[rsquo]|. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com](/nature/journal/v406/n6792/images/406188aa.0.jpg)
a, Parasegmental boundaries divide columns of wg-expressing cells (green) from columns of en-expressing cells (blue). b, 'Crisp' initial conditions: wg activated in every fourth cell, en immediately posterior. Wg and En proteins are pulsed initially in the same pattern. c, 'Degraded' initial pattern. d, Best pattern achieved with solid lines in Box 1. e, Pattern achieved with dashed lines installed. For clarity we show a single strip of twelve cells; we impose repeating boundaries, so all cells have six neighbours no matter how many are in the field. Adding rows or columns has no effect.
High resolution image and legend (43K)Using only the solid lines in Box 1a we found no such parameter sets despite extensive efforts. Most randomly chosen parameter sets caused model components to oscillate strongly or caused some components to be expressed ubiquitously while others were repressed everywhere. Figure 1d shows the pattern most resembling the target, obtained for around 1 in 3,000 randomly chosen parameter sets from the initial pattern in Fig. 1b. No parameter sets produced stable asymmetric patterns. We realized that if Wg is the only input to en and Wg is secreted symmetrically from wg-expressing cells, expression of en must be activated in all neighbours. Similarly, as Hh signalling activates wg in neighbouring cells, wg must be expressed on either side of en-expressing cells. Thus, the solid connections in Box 1a cannot suffice to explain even the most basic behaviour of the segment polarity network. There must be both active repression of en in cells anterior to the wg-expressing stripe and something that spatially biases the response of wg to Hh. There is good evidence in Drosophila for wg autoactivation16, and suggestive evidence that the Ci amino-terminal repressor fragment may inhibit en17. We incorporated these two possible remedies first (dashed lines, Box 1a). With these links installed there are many parameter sets that enable the model to reproduce the target behaviour, so many that they can be found easily by random sampling.
Each parameter set for which the model mimics Fig. 1a
we call a 'solution' to the problem posed. Among 240,000 randomly-chosen
parameter sets we found 1,192 solutions (
1 in 200). This is very frequent;
as this search involved 48 parameters, on average a random choice of parameter
value has roughly a 90% chance of being compatible with the desired behaviour
(0.948 is 1/200). This holds even though most parameters
range over several orders of magnitude. For comparison, if the model tolerated
variation in the average parameter over 10% of its 100- or 1,000-fold range
(a wildly optimistic expectation for a human-engineered electronic circuit),
random search would find only one solution in 1048 samples. Figure 1e shows the stable pattern evolved for one such
set of parameters from the pre-pattern in Fig. 1b. Clearly,
under these conditions, the network model produces a pattern comparable to
the target behaviour.
Figure 2a shows all 1,192 solutions found. Although some parameters cluster more tightly than others, none are confined to narrow sub-ranges. For each parameter, there is a solution for essentially any value. Thus, the network's ability to pass our test is intrinsic to its topology rather than to a specific quantitative tuning. There are so many diverse solutions that the notion of a globally optimal parameter set makes no biological sense. For instance, solutions for Wg diffusion rates (kMxferWG, Fig. 2a) range over three orders of magnitude, from values allowing very little Wg traffic to values for which Wg diffuses rapidly across the segment.
Figure 2: Graphic representation of 'solutions' obtained with crisp or degraded initial conditions.
![Figure 2 : Graphic representation of |[lsquo]|solutions|[rsquo]| obtained with crisp
or degraded initial conditions. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com](/nature/journal/v406/n6792/images/406188ab.0.jpg)
a, All 1,192 solutions found with crisp initial conditions.
b, Solutions found with degraded initial conditions. Black lines plot
mean and s.d. Each spoke represents the log-scale range of one parameter.
Half-lives and cooperativity coefficients are omitted for clarity. Each polygon
traces one parameter set as it intersects each spoke at the value of the indicated
parameter. Polygons of extreme red intensity earned best scores; those of
extreme blue intensity barely passed. For 
XXyy, inner
values represent potent regulation of yy by XX, outer values
the opposite. For CCID (cleavage of Ci), kPTC-HH (Ptc-Hh
binding), kMxferWG, kLmxferWG and similarly named (transfer
processes), inner boundary means slow, outer means fast.
To assess sensitivity to variation in individual parameters, we took parameter sets known to produce the desired behaviour and varied one parameter while holding all others fixed. In most cases, the model tolerates tenfold or more variation in the values of individual parameters (Fig. 3). In parameter space, abrupt transitions delineate zones within which the model behaves as desired from zones of qualitatively different behaviour. The canyons of working territory are sometimes narrow, but more often broad; the model often performs equivalently despite 100- or 1,000-fold variation in the value of some of the parameters. Thus, not only does the network topology embody many different solutions, but most solutions are highly robust to variation in individual parameter values.
Figure 3: Sensitivity of individual solutions to varying individual parameters.
Each column represents one solution, and rows are transects in which the
named parameter varies while others are held fixed. The vertical axis is the
goodness-of-fit score; lower scores are better matches. The dashed line indicates
the boundary below which we accept the match. The horizontal axis is the parameter's
log-scale range, three orders of magnitude for all. Columns one and two are
typical. The third is an unusually brittle solution: CNen
and ENcid can vary at most twofold. Column four shows the
opposite extreme, and column five is a case with two working ranges for
CIDptc and CNptc.
Aside from patterns like Fig. 1e, the model has a
complex repertoire, selected by initial conditions and parameter values. Many
randomly sampled parameter sets lead to ubiquitous expression of some subset
of model components and global repression of others. About 1 in 700 random
sets evolves the degenerate pattern in Fig. 1d for the
pre-pattern in Fig. 1b. Among the most common degenerate
patterns are a one-cell-wide wg stripe overlayed by a three-cell-wide
en strip, and a stripe of en-expressing cells surrounded by
wg-expressing cells (1 in 70 randomly chosen parameter sets each);
the former pattern results for inadequate en repression and the latter
from excessively avid wg autoactivation. Non-degenerate patterns include
en and wg expressed in the same one-cell-wide stripe (1 in
350) and en and wg expressed in overlapping three-cell-wide
bands (1 in 300). Although this is nowhere near a complete catalogue
of the model's pattern-formation repertoire, these alternate regimes are among
the 'near neighbours' of Fig. 1e in the
sense that tuning parameters across the abrupt edges of canyons in
Fig. 3 often yields these patterns.
We evaluated the model's sensitivity to initial conditions and discovered that the same stable pattern arises for a variety of input stimuli. The model has no 'wavelength'; that is, parameter values for which the model holds a four-cell-period repeat (Fig. 1e) enable it also to hold an equivalent three-, five- or arbitrary-period repeat pattern when triggered with a pre-pattern with corresponding spacing. We find that many solutions require only some initial bias towards expressing wg in one column, and en immediately posterior. Even for a very vaguely specified pre-pattern (Fig. 1c) we find solutions frequently (1 in 5,000)—still extremely high compared to the benchmark of 1 in 1048 cited above. Solutions for this case remain distributed throughout the parameter space (Fig. 2b). The model can achieve the target pattern with high frequency from initial conditions that do not include an initial pulse of wg or en ( Table 1). Clearly, the model places few absolute demands on initial conditions, and it seems likely that the evolutionary process could replace inputs relatively easily.
With our model, we reconstituted in silico an aspect of biological behaviour from a subset of the known facts, much as a biochemist might reconstitute translation in vitro. Our reconstitution is far from complete. There are many additional segment polarity genes and many inputs to them. We simulated neither cell proliferation nor rearrangement, both of which affect the real network; additional components would help to integrate patterning with morphogenesis. Many segment polarity genes function as intermediate steps between components of our model or provide outputs to downstream targets. Despite its simplicity, our model illustrates a potentially valuable benefit of the general approach. Biologists' maps of gene networks are rapidly outgrowing our ability to comprehend genetic mechanisms using human intuition alone, as shown by our initial failure. Our results reveal holes in the current understanding of segmentation: what represses en anterior to the wg-expressing stripe, and what makes Hh signalling asymmetric? We incorporated the two simplest hypotheses here, but there are hints in the literature of other candidate mechanisms. In Drosophila the hole-filling utility of models is limited because developmental geneticists will probably fill in the holes fast enough without help from models. For other organisms (like humans) models may complement more limited experimental opportunities.
More importantly, computer simulations allow biologists to explore emergent systems-level properties of gene networks. Boolean networks and random directed graphs have been used to capture the 'statistical mechanics' of genetic systems18. Such idealizations allowed the exploration of enormously complex systems and the discovery of generic properties of ensembles of randomly wired networks. Many have used similar methods to capture specific biologically realistic behaviours, including developmental pattern formation in Drosophila19. Meanwhile, the use of continuous nonlinear dynamical systems has been advocated to express cell fate determination mechanisms and the maintenance of cell states20. Until recently this approach, which we take here, faced two obstacles: a paucity of facts about specific molecular mechanisms and limited computational power for solving nonlinear models. As these constraints evaporate, realistic dynamical models, based either on mass action or stochastic kinetics, will increase in usefulness. Slack foresaw that such tools would be most useful to the extent that complex genetic circuits decompose into quasi-autonomous subsystems, that is, modules20. Our work represents such a case. In another notable example, two models have been used to express the adaptive response of the bacterial chemotactic receptor, both concurring that the mechanism is highly robust21, 22.
The most striking systems-level property we report is the robustness to parameter variation. This is not an artefact of the wiring of our model. In work to be described elsewhere, we have analysed models that include additional links and components. Our conclusions hold for all biologically grounded variants as long as they retain the core topology shown in Box 1. Why should the segment polarity mechanism be so robust? Varying parameter values is proxy for mutations of small effect, and variation in initial conditions mimics one aspect of developmental 'noise'. We are exploring how much developmental noise embryos experience, which may explain why gene networks need buffering. Alternatively, in the evolution of segmentation there may have been pressure to neutralize mutations of small effect. We originally expected the core topology to be frail and easily perturbed, and expected to achieve robustness only by adding additional complexity; we expected the reconstitution approach to tell us which architectural features confer robustness. Confounding that expectation, the simplest model that works at all emerged complete with unexpected robustness to variation in parameters and initial conditions.
Methods
Our model is a system of nonlinear ordinary differential equations, each characterizing the time-dependent change in concentration of one of the components of Box 1a in an indexed cell or cell face (see Supplementary Information for further details). All equations consist of either standard kinetic formulas or pseudo-steady-state approximations. Each generically includes three classes of additive term: a synthesis term, a first-order decay term, and zero or more terms representing transformation processes (heterodimerization or cleavage, for example) or flux between compartments (exocytosis or cell-to-cell traffic). We discretized diffusion according to cell faces: membrane-bound and extracellular molecules equilibrate at parameterized rates between adjacent faces of a cell, and extracellular molecules equilibrate at parameterized rates between adjacent faces of a cell, and extracellular molecules also exchange from one cell's faces to the apposite faces of neighbouring cells.
We ran all simulations using a prototype (versions 0.62–0.66) of Ingeneue, a custom software package that we developed to construct and analyse models such as this. Ingeneue's core function is to parse a script describing a network like that in Box 1a, convert it internally into a system of differential equations, and then numerically solve the system while monitoring the time-dependent behaviour. For the network in Box 1a, each four-cell-wide by one-cell-high 'segment' repeat unit contributes 136 coupled equations that must be solved over a simulated time interval of hundreds to thousands of minutes. This calculation takes a few seconds on a desktop computer. We quantified the model's behaviour using a customized goodness-of-fit function that assigns a scalar score to each parameter set according to how well the model governed by that set matches the desired spatial pattern of gene expression within a set amount of time (typically 3 h) and holds it stably for a longer period (typically 15 h) (see Supplementary Information). Initial conditions, specified as starting concentrations, have no enforced persistence and are the only spatially heterogenous influence. Ingeneue searches parameter space, using either random sampling or nonlinear optimization algorithms, for sets of parameters that confer on the model the desired spatial pattern formation behaviour. Ingeneue is written in Java and runs on any computer platform for which a Java Virtual Machine v.1.1.7 or better is available. Software and model files are available from the authors at http://www.ingeneue.org
We composed Box 1a from the literature on segmentation in Drosophila as follows: Wg stimulates en transcription in neighbouring cells23, 24; En promotes hh transcription25 and represses ci26; Hh binds to and sequesters Ptc27, decreasing the rate at which Ci is processed to form a repressor, CiN7517, whereas CiN75 represses wg and ptc, full-length Ci activates these genes17, 28, 29, 30; either Ci or CiN75 represses hh29; dashed lines indicate autoregulation of wg by an incompletely characterized pathway16, and a suggested repressive effect of CiN75 on en. Also, ci is basally expressed. Not shown but included in the model are transcytosis and cell-to-cell diffusion of Wg; as Wg transfers from cell to cell, Wg produced in a particular cell can activate targets in that cell.
