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NTI conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, and wrote the paper.

The author has declared that no conflicts of interest exist.

A complex hierarchy of genetic interactions converts a single-celled

The striped segmentation pattern of the Drosophila embryo is remarkably insensitive to variation (robust). Ingolia uses computational methods to show that this robustness results from specific positive feedback loops.

The network responsible for segment polarity in the

(A) Parasegments in the segment polarity pattern. The prepattern, with stripes of

(B) A simple set of rules sufficient to achieve segment polarity patterning. Cells expressing

(C) The behavior of isolated cells for parameter sets that form the segment polarity pattern. These are like the simple rules in (B), but

Many of the qualitative interactions between the components of the segment polarity network are known, but there is little quantitative information about the abundance of the components or the parameters that govern the reactions amongst them (

I asked what general features of the model yield this robustness. As defined by von Dassow et al., the task of forming the segment polarity pattern is simple. Embryos in the model begin with a prepattern composed of a repeating unit of three stripes that encompasses four rows of cells. The first stripe expresses

Thus, stable maintenance of

(A) The regulatory network used in the

(B) The regulatory network of the model developed here. The positive feedback systems are colored as in (A). The

To address this question, I asked two questions: could modeling the behavior of individual cells reproduce the overall behavior observed by von Dassow et al., and could I produce simple rules that predicted how the individual cells would behave. When I simulated the behavior of individual cells using the von Dassow et al. model, I found that individual cells in their model can adopt three different stable states of

(A) Parasegments in the segment polarity pattern during cell proliferation. During cell proliferation, each cell duplicates into two cells that initially have identical gene expression. This yields wide stripes of

(B) A simple set of rules sufficient to maintain narrow boundaries after cell proliferation. These are like the simple rules in

I began by asking if the von Dassow et al. model could be decomposed into the properties of individual cells. The simplest hypothesis is that parameters that allow individual cells to maintain their initial state of

I began by studying the properties of

In the segment polarity pattern, cells on the posterior side of the

To determine how well the isolated cell rules captured the requirements for patterning, I generated random parameter sets and tested them against the single-cell behavior rules, as well as determining whether they formed the segment polarity pattern, to see how well these correlated. Around half of randomly generated parameter sets that conform to the rules actually achieve the desired segment polarity pattern (

Random parameter sets were generated and tested for segment polarity patterning using the stripe threshold scoring scheme as described in

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Asking whether mathematical expressions can predict the behavior of single cells and the parasegment as a whole is a more stringent test of the idea that the bistability of positive feedback loops explains these stable expression states. Whether a positive feedback loop shows bistability depends on the quantitative values of its parameters. Thus, if I can predict which sets of parameters produce bistable expression of _{WG→wg}, a parameter indicating the amount of intracellular WG needed for half-maximal activation of

(A) Subnetwork responsible for _{WG}, Exo_{WG}, LMxfer_{WG},_{WG}_{EWG}_{IWG}

(B and C) Intercellular WG levels in a cell expressing _{WG→wg}, the threshold level of intercellular WG protein needed for

(D) Levels of extracellular WG signalling to a cell adjacent to two with full _{EWG→en}, the threshold level of extracellular WG signal needed to activate

(E) Steady-state levels of CN in the absence of _{CN┤en}, the threshold level needed to repress

(1) For high _{WG→wg}, the amount needed for half-maximal activation of _{WG→wg} (

(2) Transport of WG from a neighbor with high levels of _{WG→wg}. Similarly, levels of WG protein that accumulate in cells with low _{WG→wg}, but the concentrations in its non-

(3) Extracellular WG from two neighbors with high levels of _{EWG→en}, the amount of extracellular WG signal needed for half-maximal induction of _{EWG→en} indicates the amount of extracellular WG signal needed for half-maximal _{EWG→en}. Working parameter sets satisfied this constraint (

(4) The steady-state level of the CI amino-terminal fragment (CN) must be greater than K_{CN┤en}, the amount of CN needed for half-maximal repression of _{CN┤en}, the amount of CN needed for half-maximal repression of

Through such comparisons, I found that of the 0.61% of parameter sets that produced the segment polarity pattern, more than 90% were predicted to produce bistable behavior in both the

While 8.2% of all random parameter sets are consistent with the above restrictions, only 0.56% actually form the segment polarity pattern (see

The predictive value of the bistability rules is in marked contrast to the performance of the isolated cell rules, for which half the parameter sets that satisfied the rules produced the correct segment polarity pattern. I believe that the methods differ because the isolated cell rules address the dynamics by using simulations in which early expression dynamics actually occurred, at least in individual cells, rather than the steady-state comparisons of parameters used in the bistability rules. To test this possibility, I asked if I could improve the predictive value of the bistability rules by choosing different initial conditions. I focused on initial levels of CI, PTC, and CN outside the stripe of

Maintaining the narrow parasegment boundary after cell division is an important role of the segment polarity network. Even at the level of the isolated cell rules, there is a discrepancy between the behavior of the model and experimental results. Experimentally, the maintenance of

I wanted to modify the model so that it succeeded at this patterning task as well. Principally, I needed to make

This modified model can robustly form the segment polarity pattern. Taking the same approach of testing random parameter sets, I found that 9.6% could generate the segment polarity pattern. This is an 8-fold higher fraction of successful parameter sets than that seen for the von Dassow et al. model or any subsequent variants (

(A) Intercellular WG levels in a cell that expresses _{WG→wg}for each parameter set that forms the segment polarity pattern, as in

(B) Intercellular WG levels in a cell that is expressing _{WG→wg}. Parameter sets that can produce the proper pattern after proliferation, including narrow stripes of

Random parameter sets (

Rules 1 and 2 are exactly analogous to bistability rules for the original model that ensure bistability in

In addition to maintaining the initial segment polarity pattern, the modified model is also capable of producing the proper pattern after cell division. Fully 1.7% of random parameter sets yielded the desired narrow stripes of gene expression after division, showing that this feature of the modified model is also robust (see

These rules are reasonable predictors for proper behavior during cell proliferation. However, there are a substantial fraction of parameter sets that work despite breaking one or both rules, as well as many which obey them yet cannot produce the proper pattern after one round of cell division. Some of these difficulties probably result from the dynamic nature of the underlying process. As discussed above, bistability rules such as these can determine when a particular expression state is stable, but it is much harder to determine which stable state will be reached for a given initial condition. Thus, it is possible to predict when a parameter set will be able to maintain the final postproliferation pattern, but it is much harder to determine when it will reach this pattern from the expression state immediately following proliferation. This does not explain why there are parameter sets that do not obey rules 5 and 6 but nonetheless give narrow stripes of gene expression after cell division. Those parameter sets expose limits in the approximations used to develop the cell bistability rules. There may be small but important interactions between different feedback loops within a single cell, or perhaps some aspect of intercellular signaling is more complicated than the simple binary model employed in the bistability rules.

I have shown that individual cells in the segment polarity model can adopt three distinct expression states, influenced by signals from their neighbors. I have also presented evidence that positive feedback in the model produces these states. The importance of autoregulation in establishing distinct expression states has been recognized in this system before (

The approach I have taken can be generally applied to models of complicated genetic or biochemical networks. I isolated small subnetworks, chosen to be maximally insulated from the rest of the system, and studied their behavior in isolation. This let us understand the principles that allow the entire network to function. I verified this understanding by creating tests for the behavior of the subnetworks and showing that these were powerful predictive tools for the performance of the entire network. This sort of decomposition is also useful in combination with quantitative phenomenological descriptions of subnetwork behaviors. Recent experimental studies provide such descriptions for a number of biological systems, including vertebrate homologues of the

The robustness of the segment polarity network is a result of the fact that the desired pattern is a stable steady state. In a system of ordinary differential equations, such as the models described here, such states correspond to stable fixed points. These are generic features of such systems; small changes in parameters or initial conditions will not change them qualitatively. This can be seen in the bistability rules I developed. They are inequality constraints, so they carve out a volume of parameter space in which parameter sets can maintain the segment polarity pattern. In our analysis, I focused on robustness against changing parameters, which correspond to genetic alterations that change quantitative values of reaction parameters. In the real world, stochastic and environmental perturbations in the system may play at least as large a role.

One important question is the extent to which the behavior of a network is determined by its topology, as opposed to quantitative details. The network topology is just the set of interactions in the network, along with their signs. This information is accessible to standard, qualitative biological experiments. Topology limits the possible behaviors of a regulatory network. Positive feedback, which is a topological property, is necessary for multiple stable states (

These examples show how changes in the quantitative details of a regulatory network can result in qualitatively different behaviors. This could explain how pattern formation can be evolvable; mutations which cause large shifts in a critical parameter could cause a network to form a different pattern corresponding to a new stable state. The altered pattern would still correspond to a stable fixed point, so it would also be robust against various kinds of perturbations. This offers a mechanism that could produce new patterns without nonfunctional intermediates and without events such as the creation of a new protein–protein interaction.

Isolated cell rules were tested by simulations in which the dynamics of an individual cell were modeled using the same equations that govern each cell in the segment for the full segment polarity network.

Since WG protein diffused between cells as well as moving into and out of a given cell, it was important to account for the diffusion of WG even in isolated cell simulations. The level of _{i}_{i}_{i,j}. Extracellular WG can exchange between faces of the same cell and between opposing faces of adjacent cells.

The parameters _{IWG}_{EWG}_{out,WG}_{across,WG}_{around,WG}_{in,WG}_{i}_{i,j}_{i}s_{on}_{off}_{in,WG}^{−1}_{EWG}_{i,j}^{kt}_{i,j} as the amount of WG protein on neighboring cells for the isolated cell simulations.

To verify bistability of _{on,0}_{off,0}_{i,j}_{on}_{off}_{on,i+1} and _{off,i+1} to compute the levels of extracellular WG protein for the next iteration. This process quickly converged, and I took the resulting _{on}_{off}_{on}_{off}

I then used the same levels of extracellular WG protein, computed from _{on}_{off}

These equations make repeated use of a particular equation form representing saturable and cooperative action of a protein, for instance as a transcriptional activator. In general, the amount of activation, Φ, as a function of the concentration of activator,

Here, _{EWG→en}, which indicates the amount of extracellular WG needed for half-maximal activation, and by ν_{EWG→en}, which determines how cooperative the activation is.

I designed parameter rules for bistability by analyzing different subnetworks in the model and solving for steady states consistent with bistability from positive feedback. I solved for the stationary state of the _{act}_{rep}

The affinity and cooperativity parameters for each Φ have been suppressed for clarity. The parameters _{ci}_{ptc}_{Ptc}_{Ci}_{CN}_{0} is an affinity parameter for the cleavage of CI by PTC. To find the stationary state, I solve for the simultaneous zero of all five equations. Two variables, _{rep}, was compared to K_{CN┤en}, the amount needed for half-maximal repression of

The levels of CI and CN were also used to compute their influence on

The only parameters in this expression are the affinity and cooperativity parameters for each Φ.

Levels of _{i}_{i}

In addition to affinity and cooperativity parameters for each Φ, and _{wg}_{CI→wg} and α_{WG→wg}, which determine the relative strengths of CI/CN and WG influences on _{i}_{WG→wg}, then Φ(_{i}

Bistability requires that intracellular WG levels in a _{w=1}=_{i}_{i}_{on}=_{i}_{i}_{w=1}) by setting _{on}=_{i}_{i}_{on}_{WG→wg}, meaning that the level of intracellular WG is sufficient to maintain _{on} to compute the amount of intracellular WG in a cell next to the _{nbr}_{i+1}(_{i}_{on},_{i+1}=0), and found _{off}_{i+1}(_{i+1}=_{nbr}) and _{off}=_{i+1}(_{i+1}=_{off}). I then verified that _{nbr}+_{off}<_{WG→wg}, meaning that the sum of intracellular WG transported into a _{on,j} and _{off,j} in the same manner as _{on} and _{off}, respectively. These are used to ensure that the level of extracellular WG signal received by a cell in the _{EWG→en} (parameter rule 3).

The modified initial conditions were generated by solving for the steady state of the CI and PTC subnetwork as described above. This yielded steady-state values _{rep}, _{act},

The modified initial conditions also used steady-state levels of intracellular and extracellular WG protein. The steady-state _{i} and _{i,j} values were computed as described above under the assumption of a single column of cells with maximal

The equations governing the modified model were similar in form to those in the original model. In addition to using the functional form Φ(_{r}_{a}

Again, _{0}_{r}

In addition to the dynamic variables described above, levels of _{Nbr} or _{Nbr} indicates the sum of extracellular WG or HH on neighboring cells, respectively; this is equivalent to the average without a normalization for the number of cells.

Initial conditions were

Steady-state levels _{on}_{off} were computed similarly to the way described for the original model. I assumed maximal _{Nbr}=2, in computing the steady-state levels _{act} and _{rep}. As there was no intercellular transport of WG in the modified model, I needed to worry only about basal and activated _{on} and _{off} to K_{WG→wg}.

I computed _{w=1} for _{Nbr=2} to account for WG signaling in _{S=0} using _{Nbr}=2_{w=1} to represent maximal WG signaling from two neighbors and _{Nbr})·(1−Φ(_{off} using the steady-state equation _{off} and _{w=1} to find _{on} in the _{EN┤slp} to ensure that steady-state levels of EN were sufficient to repress

Similarly, I found _{S=1} using the steady-state _{on} using the steady-state _{on} was then used to find _{off}, and I required that _{off}<_{EN┤slp}. This ensured that repressed levels of

To test that _{on} as described before. I also computed _{act} and _{rep} using _{Nbr}=0, representing a loss of HH signaling. I then used _{on} and the new _{act} and _{rep} to find _{H=0} and _{H=0} with the steady state _{on→off} and _{on→off} using the steady state _{act} and _{rep}, and _{H=0}. Finally, I verified that _{on→off}<_{WG→wg}, which ensure that

To check whether _{on} and _{off} as described above. I found _{off} in the same way in which I found _{off} and used _{Nbr,off}=6_{off}. I used this new level of WG signaling to find _{on→off} with the steady state _{off→on} with the steady-state _{off→on}>_{SLP┤en}, ensuring that the unrepressed level of

Additional background and explanation of bistability in gene expression.

(109 KB PDF).

The FlyBase (

I thank Andrew Murray for advice and helpful discussions. I also appreciate critical comments on the manuscript from Daniel Fisher, Steve Altschuler, Lani Wu, Scott Schuyler, and all the members of the Murray laboratory as well as the anonymous reviewers. This work was supported by a predoctoral fellowship from the Howard Hughes Medical Institute. It was conducted in the laboratory of Andrew Murray, who is supported by the National Institutes Health.

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Cubitus Interruptus

Cubitus Interruptus amino-terminal fragment

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wingless.