Research ArticleCOMPUTATIONAL BIOLOGY

Error Minimization in Lateral Inhibition Circuits

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Science Signaling  06 Jul 2010:
Vol. 3, Issue 129, pp. ra51
DOI: 10.1126/scisignal.2000857

Abstract

The pattern of the sensory bristles in the fruit fly Drosophila is remarkably reproducible. Each bristle arises from a sensory organ precursor (SOP) cell that is selected, through a lateral inhibition process, from a cluster of proneural cells. Although this process is well characterized, the mechanism ensuring its robustness remains obscure. Using probabilistic modeling, we defined the sources of error in SOP selection and examined how they depend on the underlying molecular circuit. We found that rapid inhibition of the neural differentiation of nonselected cells, coupled with high cell-to-cell variability in the timing of selection, is crucial for accurate SOP selection. Cell-autonomous interactions (cis interactions) between the Notch receptor and its ligands Delta or Serrate facilitate accurate SOP selection by shortening the effective delay between the time when the inhibitory signal is initiated in one cell and the time when it acts on neighboring cells, suggesting that selection relies on competition between cis and trans interactions of Notch with its ligands. The cis interaction model predicts that the increase in ectopic SOP selections observed with reduced Notch abundance can be compensated for by reducing the abundance of the Notch ligands Delta and Serrate. We validated this prediction experimentally by quantifying the frequency of ectopic bristles in flies carrying heterozygous null mutations of Notch, Delta, or Serrate or combinations of these alleles. We propose that susceptibility to errors distinguishes seemingly equivalent designs of developmental circuits regulating pattern formation.

Introduction

Development of metazoan organs frequently involves the selection of a single cell from a cluster of cells that have an equivalent potential for differentiation toward a specific cell fate. Lateral inhibition, whereby the selected cell blocks the ability of its neighbors to differentiate, plays a key role in such processes. The sensory bristles of the fruit fly Drosophila arise from specialized cells, the sensory organ precursors (SOPs), that are specified during development (14). At the onset of neural differentiation, a group of cells (the “proneural cluster”) begins to produce the proneural transcriptional activators Achaete (ac) and Scute (sc) (Fig. 1A). A single cell from this cluster maintains abundant quantities of these factors and will become an SOP (Fig. 1B). This selected cell inhibits its neighbors by producing large amounts of the Notch ligands Delta and Serrate, which bind to the Notch receptor on adjacent cells, thereby triggering generation of the activated form of Notch [the free intracellular Notch domain (NID)]. NID induces transcription of genes in the Enhancer of Split [E(spl)] repressor complex, whose products antagonize transcription of the proneural activators ac and sc and thereby prevent differentiation into an SOP. The fidelity of the selected SOP fate is further ensured by the production of E(spl) antagonists, such as Senseless (5), and the inhibition of autonomous Notch activity by its binding with Delta and Serrate in cis, within the same cell (68). This process of SOP selection occurs repeatedly at different stages of fly nervous system development, generating different sensory organs, such as the large bristles (macrochaetes) and the small bristles (microchaetes) of the fly’s thorax (Fig. 1C).

Fig. 1

The fly sensory bristles are formed by a lateral inhibition process. (A) The appearance of the proneural cluster (gray cells) is the initial step in the selection of an SOP. (B) Later in the developmental process, a single cell from the proneural cluster (A) is selected as an SOP (blue) and, via a lateral inhibition process, represses the neural differential potential of its neighbors and directs them to a non-SOP epidermal fate (red). (C) Light microscopy image of thoracic bristles in a control fly. An example of the scutum small bristles (microchaetes) is marked with a gray arrow. The large bristles (macrochaetes) of the fly’s scutellum are marked with arrowheads (aSC, up, and pSC, down). Rarely, large bristle duplication is observed in control flies; shown are two adjacent macrochaetes (arrow) formed at a single aSC position.

In wild-type flies, SOP selection is highly accurate. The reproducible number and pattern of the large sensory bristles served as an experimental paradigm for illustrating the concept of developmental canalization (the preservation of phenotype despite variation in genotype and environment) (9). These bristles are arranged in a characteristic manner, with two large bristles positioned on each side of the fly’s scutellum (the posterior part of the thorax) (Fig. 1C). Bristle duplication, in which two closely spaced bristles are found at positions typically associated with a single bristle, is observed in <1% of wild-type flies (9, 10) (Fig. 1C); loss of one of the bristles is even less frequent and is hardly observed in wild-type flies. However, error frequency increases when components of the lateral inhibition circuit are perturbed; for example, duplicated bristles are not uncommon in flies heterozygous for a null allele of the Notch receptor, and a loss-of-bristle phenotype is occasionally observed in flies expressing a gain-of-function Notch allele (11).

Previous mathematical analyses have confirmed that lateral inhibition circuits can amplify small differences between adjacent cells, and indicated that activity of the Notch-Delta circuit can select a single SOP from a cluster of proneural cells (1214). The possible sources of error in this selection process, however, have not been analyzed, nor is it known how error frequency is influenced by different possible designs of the lateral inhibition circuit. Here, we use probabilistic modeling to quantify the errors in processes mediated by lateral inhibition, define the sources of selection errors, and examine the molecular designs that minimize these errors during SOP selection.

Results

Including refractoriness and a delay in inhibition in a model of lateral inhibition

We first considered a simplified model of two interacting cells, each producing a single proneural component (M) that inhibits production of its counterpart in the adjacent cell (Fig. 2A and Materials and Methods). The two variables M1 and M2 are produced and degraded over time and exert lateral inhibition so that M1 inhibits M2 production, and M2 inhibits M1 production. For simplicity, we assume a threshold for initiation of inhibition, so that there is no inhibitory effect until a certain amount of inhibitor has accumulated, and linear degradation of M (15).

Fig. 2

Sources of errors in SOP selection. (A) A single-component model of lateral inhibition. The model consists of a single-component M in two interacting cells. M1 and M2 (M in cell 1 and cell 2, respectively) increase and, upon reaching a certain threshold, begin to laterally inhibit their counterpart (see Materials and Methods for mathematical description). (B) Simulated selection. M accumulates in both cells (solid and dashed lines) until the inhibitory threshold is reached in one of the cells, immediately repressing M production in the neighboring cell (see Materials and Methods). The y axis shows normalized M abundance. (C) A single-component model incorporating a delay (see Materials and Methods). (D) Failed selection in the presence of a 10-min delay. Two cells approach kin, the inhibitory threshold, at similar times, after which the cells could (i) oscillate (blue, kr = 30 molecules), (ii) be irreversibly inhibited (red, kr = 30 molecules), or (iii) become refractory to mutual inhibition (black, kr = 20 molecules). (E) Time-lapse imaging of SOP selection in live pupal disc. All cells are labeled with green fluorescent protein (green); Notch activation is visualized with the E(spl)mα-DsRed.T4-NLS reporter (red). SOP (marked by arrowheads) is identified by increased reporter abundance in all adjacent cells (see section S10 and fig. S4). Times are relative to head eversion (HE) ~12 hours after pupal formation, as indicated. (F) A rare event where two adjacent cells are selected (arrowheads).

Production of M1 and M2 is initiated at time t = 0, after which they begin to accumulate. Because of small stochastic differences, either M1 or M2 will reach the threshold for initiating inhibition first and will begin to inhibit production of its counterpart. If inhibition is instantaneous, repression of neural differentiation of the neighboring cell is guaranteed, leading to the unequivocal selection of the cell that produces threshold amounts of M first (say the cell that produces M1) as the SOP (“Proper selection,” Fig. 2B). In reality, however, inhibition is mediated by biochemical reactions and thus inherently takes some finite amount of time. Consequently, M2 concentrations could reach the threshold for mediating repression of M1 before M2 production was inhibited (Fig. 2C). In this case, errors in selection can occur, and their types and frequency depend on the dynamics of M after the repression threshold had been crossed.

Several scenarios can be envisioned (Fig. 2D). In one scenario, mutual inhibition leads to a transient reduction in the concentrations of both M1 and M2, thereby relieving the mutual inhibition and permitting additional attempts for selection. In this case, oscillations will ensue until one of the cells crosses the threshold early enough to fully inhibit its counterpart (“Oscillation,” Fig. 2D). If iterations are allowed to proceed indefinitely, a single cell will eventually be selected, leading to an error-free process. Selection is not guaranteed, however, within a finite time window. In a second scenario, inhibition is irreversible, so that once production of M is repressed, it can no longer be activated (“No relief,” Fig. 2D). In this case, mutual inhibition will lead to an error in which neither cell is selected. In the SOP example, this would lead to the lack of bristle formation in the appropriate locale. And, in a final scenario, cells that have passed the threshold could become refractory to inhibitory signals, in which case high M will be retained in both cells (“Refractoriness,” Fig. 2D). Here, mutual threshold crossing will result in an error of the duplicated bristle type.

The primary error observed in wild-type flies is duplicated bristles. This suggests that refractoriness may play a role in the selection process. Duplication, however, could occur at later stages of development, after SOP selection through lateral inhibition. To determine whether cells become refractory during the selection process itself, we monitored the in vivo selection of the SOP of the small bristles (16, 17) (Fig. 2E). We detected cases in which adjacent cells resisted strong activation of Notch by a neighbor, leading to selection of two SOPs (Fig. 2F). This suggests that cells can become refractory to Notch signaling at about the time lateral inhibition is initiated, arguing for a role of refractoriness in the lateral inhibition dynamics.

A probabilistic model of calculating error rates in processes mediated by lateral inhibition

We next extended the model to include two additional properties. First, we introduced a delay from the time at which one of the cells has crossed the inhibition threshold (that is, produced enough M to inhibit its neighbors) until the differentiation potential of its neighbor is inhibited. Second, we considered the possibility that the selected cell becomes refractory to inhibition by its neighbors (18). In addition, because a cell that is not inhibited becomes an SOP by default [as observed, for example, with proneural clusters in which all cells lack the Notch ligands and therefore cannot initiate lateral inhibition but maintain abundant levels of the ac/sc factors and differentiate toward the SOP fate (19)], we assume that inhibition must take place within a finite time window and that a cell that is not inhibited within this time window will necessarily be selected. Thus, an error of the duplicated SOP type is predicted either when two cells cross the threshold before being inhibited or when inhibition is not initiated within the available time.

Using probabilistic analysis, we estimated the rate of selection errors in this model (see section S1 for full analysis). In brief, we considered a field of n interacting cells and summarized the independent dynamics of each cell i using the following three different times, measured from the onset of M accumulation. In our model, the ith cell begins to produce an inhibitory signal at tin, the “inhibition time”; this cell becomes insensitive to inhibition by its neighbor at the “refractory time,” tr; finally, the cell becomes irreversibly selected as an SOP at the “selection time,” ts. In our model, these times are chosen at random from distributions whose means characterize the typical dynamics of SOP selection, namely, the mean time it takes until M reaches the respective threshold, whereas their width captures cell-to-cell variability (noise) of these times. We further assumed that the inhibitory interaction between the cells is characterized by two time delays, measured from the initiation of the inhibitory signal in one cell until its neighbors are prevented from producing an inhibitory signal (the “inhibition delay,” τin), or prevented from being selected (the “selection delay,” τs).

Using these parameters, proper selection of a single cell during a single iteration requires sufficient time difference between the onset of inhibitory signal production by the first two cells to do so: Δt>τeff(1)

with Δt=t2int1in. During this time period (and only during this time period), only the first cell to initiate production of the inhibitory signal can inhibit its neighbors. Δt needs to exceed the effective delay time, τeff, which constitutes the effective delay between the times at which inhibitory signal is produced by one cell until it affects the neighboring cells, to ensure selection of this cell. The precise definition of τeff is given below.

When cells are equivalent, both t1in and t2in in our model are chosen from the same distribution, PAT(t), which we call the “arrival time” distribution (Fig. 3, A and B). Approximating this distribution by a square-like function of some width, α (α is given by the standard-deviation of the distribution, divided by its mean), we can evaluate the error rate as a function of τeff and α (section S2). This amounts to calculating the probability that the inhibition times t1in and t2in [picked from the arrival time distributions PAT(t)] will be either too close (so that condition 1 does not hold) or too long (so that no inhibition will be initiated during the available time). We find that error rate increases with the effective delay τeff and decreases with the cell-to-cell variability α (Fig. 3C). For any given τeff, the minimal error rate at which ectopic SOPs are produced is obtained when variability is maximal (α→1, corresponding to a purely stochastic process; section S3). For example, the minimal error rate, PER, for a cluster of four cells (n = 4) is given by PER1(1τeff2T)4 (2)

Fig. 3

Predicting the selection errors in the lateral inhibition process. (A) Stochastic accumulation of M. Ten different instances of M accumulation, which differ by the choice of stochastic noise, are shown in gray, with one of them marked in black. (B) The arrival time probability distribution function, PAT(t), for the system in (A). Marked are mean arrival time, <t> (dashed vertical line), noise level, α<t> (red), time window, T (blue), and inhibition delay τ (gray). (C) Dependence of error rate on noise and delay in inhibition. Minimal error rate (eq. S2.7, section S2), for a cluster of n = 4 cells and T = 360 min, as a function of inhibition delay and noise. Noise, α, is defined as the normalized width of PAT(t). Error rate of 1% is marked by the black line. The dashed line marks the maximal value of the effective inhibition delay with corresponding 1% (or less) error rate. Note the log scale for the error rate (color bar).

The error frequency, PER, given by Eq. 2 depends on the developmental time window T, which restricts the average and variance of the arrival time distribution PAT(t). Distributions that are too wide relative to T, for example, will result in cases where none of the cells initiate lateral inhibition before irreversible commitment to the SOP fate (Fig. 3B).

Because our model is general, Eq. 2 should hold for any molecular implementation of a lateral inhibition circuit that selects a single cell out of a cluster of equivalent cells. Moreover, even if the cells are not equivalent, but are subject to some preexisting differentiation pattern that biases selection toward specific cells in the cluster, the same qualitative dependence on time delay is predicted (section S4 and fig. S1).

The effective time delay

The critical parameter controlling the error rate is the effective time delay, τeff. As noted, τeff, which corresponds to the time it takes for the cell sending an inhibitory signal to actually inhibit the adjacent cells, depends on the cumulative contribution of the different delays in the system. If cells become refractory shortly after crossing the repression threshold [(trtin) < τs], as indicated by our experimental observation in Fig. 2F, then τeff is given by the sum of the delay until selection, τs, and the time difference between the onsets of inhibition and refractoriness tintr:τeff=τs+(tintr)(3)

In this case, only a single iteration is possible. If the time difference between the onset of inhibitory signal production by the first two cells to do so is sufficiently large (condition 1 holds), a proper selection of only one cell is ensured. Otherwise, two cells will be selected, leading to an error of the duplicated bristle type.

The same formalism also applies when refractoriness occurs with a delay relative to the onset of inhibition [τs < (trtin)]. The effective time delay, τeff, in this case is given by the smaller of the following two parameters: the inhibition delay τin and the delay between the onset of refractoriness and inhibition (trtin). In this case, several iterations are possible. Our error rate calculations still describe the probability of successful selection in a single iteration, but this error will be reduced with successive iterations. When the developmental time window is finite, only a limited number of oscillations are possible, and although the error rate is reduced relative to the one-trial scenario, the improvement offered by successive iterations is limited (see section S5 and fig. S2).

Requirement of a rapid effective delay to explain the accuracy of SOP selection

To study the possible implications of Eq. 2 for selection in vivo, we analyzed errors in the selection of the four scutellar bristles on the fly thorax. The SOPs that give rise to these bristles are defined during the larval stage. Each bristle arises from one SOP, which is selected from a proneural cluster of three to seven equivalent cells found at different positions on the wing imaginal disc (20). The location of this small cluster depends on an intricate system of long- and short-range patterning signals (21, 22), but the final selection of a single SOP is likely to reflect amplification of small stochastic differences between the cells by a Notch-dependent lateral inhibition circuit.

In wild-type flies, the number of scutellar bristles is highly reproducible, but duplicated bristles are occasionally observed. We quantified the frequency of this error in flies from various wild-type backgrounds and found ~1% incidence of duplicated anterior scutellar (aSC) bristles. We did not observe any instances of a missing bristle (n = 114). Moreover, we only observed duplications for the aSC bristle, but none for the posterior scutellar (pSC) bristle. Equation 2 suggests that this difference in error rate might be due to a shorter developmental time window, T, for the aSC. This parameter is difficult to measure, but a reasonable estimate is provided by the measured variability in the time of the appearance of the corresponding SOP during larval development. Indeed, this time variance is 18 hours for the pSC but only 6 hours for the aSC (23), consistent with a shorter developmental time window, and consequently, a higher error rate, of the latter. Using this 6-hour estimate for the developmental time window, T = 360 min in Eq. 2, we conclude that an effective delay τeff of <1 min is necessary to enable an error frequency of ~1% (Fig. 3C). Maintaining a low error rate thus requires rapid inhibition.

Long delays in lateral inhibition circuits that rely on transcriptional feedback

What are the implications of these results for the lateral inhibition circuit underlying SOP selection? As noted above, the effective delay depends on the different delays in the system, τs or τin, and on the period between the onsets of inhibition and refractoriness, tintr. We thus wished to estimate these parameters in specific models of the lateral inhibition circuits underlying SOP selection. In particular, we wish to distinguish between models that rely only on trans interactions between Notch and its ligands Delta and Serrate and models that also incorporate the cis interactions between Notch and its ligands.

The key players in the lateral inhibition circuit are Notch and its activating ligands, Delta and Serrate. In the simplest model, lateral inhibition is initiated once Delta or Serrate accumulates to some threshold amount (Fig. 4A and section S6) and binds to the Notch receptor on adjacent cells to generate the NID and thereby induce the expression of E(spl). Accumulating E(spl) inhibits the production of ac and sc, thereby preventing the cells from becoming SOPs, and also the production and activity of the Notch ligands Delta and Serrate, whose production depends on these transcription factors. Thus, both the selection delay τs and the inhibition delay τin depend on the accumulation of E(spl) (Fig. 4C and section S6). Accumulation of E(spl) requires transcription and translation and is thus unlikely to culminate in a manner of minutes.

Fig. 4

Notch-Delta cis interaction–based model facilitates rapid signaling and low error rate. (A) Transcription-based feedback model. Refractoriness to inhibition is caused by abundant Senseless (sens). Delta and Serrate (Ser) (blue) elicits inhibition of lateral cells (see section S6). (B) Cis interaction–based model. Lateral inhibition and refractoriness are initiated when the Notch ligands Delta and Serrate are sufficiently abundant to produce lateral inhibition and autorefractoriness (see section S7). (C) Calculated effective inhibition delays in the feedback- and cis-based models (see sections S6 and S7). Without cis interaction, the effective inhibition delay corresponds to the time required to accumulate sufficient E(spl) to repress ac/sc (upper purple arrow). With dominant cis interactions, the effective inhibition delay corresponds to the time required to accumulate sufficient NID to ensure eventual inhibition of SOP fate (lower purple arrow). (D) Selection dynamics in cis-based model. Depicted are the normalized protein abundances in selected and repressed cells (shown as dashed and solid lines, respectively, when the patterns of protein accumulation diverge). Linear accumulation of Delta-Serrate (blue) results in the depletion of Notch (through cis interactions, red) at t ~ 130 min. Rapid accumulation of NID in the neighboring cell follows (brown), leading to E(spl) accumulation (black) and a reduction in Delta-Serrate abundance. The two cells arrive at the threshold 1 min apart. See section S7 for model equations and parameters.

The second contribution to the effective time delay is tintr, the delay between the onset of inhibition and the onset of refractoriness in the same cell. This parameter, which depends on the precise process underlying refractoriness, is more difficult to estimate. In the simplest view, refractoriness is achieved through the production of antagonists, such as Senseless, that limit the ability of Notch signaling to repress ac and sc production and thereby stabilize SOP selection (Fig. 4A and section S6). Delta and Senseless are both transcriptionally induced by ac and sc; thus, we assume that tin is similar to tr. Under these conditions, the effective time delay will be determined by the long delays τs and τin.

In principle, the effective time delay could be minimized by tuning the onset of inhibition and refractoriness such that trtin = τs. We do not favor this possibility, however, for considerations of robustness; in the absence of additional regulatory mechanisms, such a fine-tuned mechanism will be sensitive to any changes that will alter the individual parameters, such as gene dosage or stochastic fluctuations.

Implications of cis inhibition for rapid signaling

Can the effective delay be shortened without relying on fine-tuning the selection delay with tintr? For this to happen, the selection delay τs must be independent of the lengthy processes of transcription and translation. This requires that inhibition of the SOP fate must be ensured before the production of E(spl). In addition, to eliminate possible contribution of tintr to the effective delay, refractoriness must be coupled to the initiation of Notch signaling (so that tin = tr). As we show below, cell-autonomous interactions between Notch and its ligands can provide both of these properties and in this way facilitate rapid inhibition and accurate selection.

The ability of Notch and its ligands to interact in cis (within the same cell) without generating NID is well established (68, 2427), but the contribution of such interactions to in vivo SOP selection is still debated. To see how cis interactions can facilitate rapid inhibition (and shorten τs), we hypothesize that the cis interaction between Notch and its ligands predominates over their transcellular interactions. Under these circumstances, Delta and Serrate accumulate gradually within each cell, but do not activate Notch in adjacent cells as long as intracellular Notch is available for cis interaction with cytoplasmic ligand. Transactivation is initiated only when free Notch is depleted by being endocytosed or saturated with ligand, so that excess ligand becomes available. This implies that cells that initiate lateral signaling are depleted of free Notch that can itself be activated in trans. Thus, cells that initiate lateral signaling are refractory to signals coming from their neighbors. Refractoriness is thus directly coupled to the initiation of inhibition, so that tin = tr (Fig. 4B and section S7).

What about the selection delay, τs? The critical distinction of the cis-dominated model compared to a model that lacks cis interactions is that refractoriness is implemented at the level of the Notch receptor, so that the refractory cell is unable to sense ligands from neighboring cells. This refractoriness, however, does not prevent inhibition of the SOP fate by activated Notch (NID) already present in the cell. Consequently, here, the selection delay (τs) represents the time needed to accumulate sufficient NID to ensure the eventual inhibition of ac and sc by E(spl) induction (Fig. 4C). However, this selection delay, τs, is controlled by rapid protein-protein interactions. For example, assuming that small amounts of NID are required to induce E(spl) transcription (28, 29), conceivably at ~1 nM, and a total concentration of Notch ~1 μM, we obtain a delay in the range of seconds, a time short enough to ensure efficient selection (Fig. 4C and section S7).

The full selection dynamics is illustrated by numerical simulations (Fig. 4D). Here, we follow two cells that approach the threshold at similar times (t1in = 129 min versus t2in = 130 min). As Delta and Serrate begin to accumulate, no trans signal is generated, but the cis interaction leads to the gradual depletion of free Notch. The dynamics of the two cells begin to differ only when free Notch is eliminated from the first cell. At this point, the Notch ligands begin to activate Notch in trans, leading to the production of NID in the second cell. NID production is transient, because free Notch is shortly eliminated from the second cell as well. However, the initial pulse of activated NID suffices to ensure the eventual permanent inhibition of ac and sc. Specifically, after this pulse, E(spl) is transcriptionally induced and its production leads, after some delay, to the inhibition of Delta and Serrate production. The ongoing degradation of Delta and Serrate enables the reaccumulation of free Notch, as its endocytosis is reduced. Once free Notch becomes available again, it enables a second sustained wave of signaling, ensuring the full repression of ac and sc and, consequently, the potential for this cell to become an SOP.

Experimental evidence supporting the cis inhibition model for SOP selection

The cis-based model assigns Notch and its ligands a dual role: They mediate not only lateral signaling but also cis inhibition. This is in contrast to the more standard description of lateral inhibition (such as the feedback model described above), in which Notch and its ligands facilitate only the transmission of the inhibitory lateral signal. Consequently, the two models predict different patterns of genetic interaction between Notch and its ligands. The feedback model predicts that reducing the abundance of either Notch or its ligands (for instance, in flies heterozygous for null alleles of Notch or its ligands Delta and Serrate) will result in a similar phenotype and that the combination of the two (for instance, flies heterozygous for both Notch and its ligands) will be synergistic, displaying a stronger phenotype than either alone. In contrast, the cis interaction model predicts compensating interactions. This is because most of the dynamics is dominated by the interaction in cis, where Notch and its ligand exert opposite effects: Notch abundance defines a threshold that needs to be surpassed by that of its same-cell ligands to initiate lateral inhibition (Fig. 3D). Consequently, it is predicted that heterozygosity for Notch will be compensated for by reducing the abundance of Delta or Serrate (section S8). Such compensating interactions have been observed previously in other contexts of Notch signaling and were among the first indications for cis interactions between Notch and its ligands (30). However, the relevance of this interaction in the context of SOP selection remains controversial.

To examine this, we counted error frequency in the adult fly, focusing on the scutellar bristles. We first created triple Notch-Delta-Serrate heterozygous flies. As predicted by the cis interaction model, the rate of duplicated selections was reduced from 28% in Notch heterozygotes (n = 54) to 0% in the triple heterozygotes (n = 94) (Fig. 5A). Compensation for Notch phenotype was also observed with Notch-Serrate heterozygous flies [with 13% duplicated selection (n = 94)].

Fig. 5

The frequency of bristle duplication in heterozygotic flies is consistent with predictions of the cis interaction–based model. (A) Measured error rates for ectopic aSC bristle (in percent) for flies heterozygous for Notch (N−/+), Notch and Delta (D−/+;N−/+), Notch and Serrate (S−/+;N−/+), and Notch, Delta, and Serrate Notch (D−/+;S−/+;N−/+). Error rate reduction for S−/+;N−/+ and D−/+;S−/+;N−/+ with respect to N−/+ is significant (*P = 0.02, **P < 10−7, Fisher’s exact text). All heterozygote alleles were crossed to the same wild-type background (yw) before measurement of error rate. See table S1 for numerical number of flies counted and details of background. (B) Predicted effective inhibition delays of wild-type flies (Cont.) and for flies heterozygous for Delta (D−/+), Serrate (S−/+), Notch (N−/+), or some combination of these. Predictions are based on simulation of the system in Fig. 4B, with production rate of the heterozygous components reduced by half (see sections S7 and S8). (C) Predicted mean selection times of flies of the denoted phenotypes. Predictions are based on simulation of the system in Fig. 4B, with production rate of the different heterozygous component reduced by half (see sections S7 and S8). (D) Correlation between measured (y axis) and predicted (x axis) error rates for duplicated aSC bristles. The error bars on the y axis represent estimated variance in the measured error rate due to low number sampling. The Pearson correlation is R2 = ~0.83 (P < 0.01).

In contrast, the Notch-Delta heterozygous flies did not show compensation [with 26% duplicated selection (n = 66)] (Fig. 5A). The difference between Delta and Serrate can be explained by the cis interaction model if we assume that the relative contribution of Delta and Serrate to the trans interaction differs. If Delta dominates over Serrate in the trans interaction, the net effect of reducing Delta abundance will be to increase, rather than decrease, the effective inhibition delay, leading to a higher error rate (Fig. 5B). However, lowering Serrate abundance will reduce the cis interaction without altering trans signaling, leading to a shorter effective inhibition delay and a lower error rate (Fig. 5B). Consistent with this model, proper SOP selection is observed within a Serrate-null clone (19), suggesting that Delta can function effectively both in cis and in trans, whereas no Notch activation is observed within Delta-null clones (19).

The cis interaction model can also predict the quantitative error values for different combinations of the Notch, Delta, and Serrate heterozygotes. This is done by calculating two parameters: the change in the inhibition delay, τeff (Fig. 5B), and the mean selection time, 〈t〉 (Fig. 5C). In contrast to the prediction of compensating interaction between Notch and its ligand, which is independent of parameters, the quantitative error values predicted depend on the values of the in vivo parameters, which are not known. Still, we can identify parameter values for which the quantitative computational predictions are in good agreement with the experimental measurements (R2 = ~0.83, P < 0.01) (Fig. 5D and table S1). One case that is not explained by the model is that of the triple Notch-Delta-Serrate heterozygote flies, which displayed a zero error rate, significantly below model prediction (~10%). This may be indicative of genetic modifiers within the specific background of the Delta-Serrate double-heterozygote allele (31).

Discussion

Here, we analyzed the sources of error in the lateral inhibition process and examined their implications for the in vivo molecular design of such circuits. Using probabilistic modeling, we quantified the rate of selection errors on the basis of two general parameters: the signaling delay and the cell-to-cell variability, or noise. We find that the delay in lateral inhibition plays a key role in determining error frequency: The shorter the delay, the better the accuracy of the selection process. We further show that a model of the lateral inhibition circuit based on the cis interaction between Notch and its ligands shortens the delay and thus improves the selection accuracy. We propose that because, in general, evolution favors molecular circuitry that minimizes errors in patterning, the selection process is likely to be based on cis inhibition. We provide experimental evidence to substantiate this proposal by analyzing the change in error rate in flies with altered gene dosages of Notch, Delta, and Serrate.

The inhibitory interaction between Notch and its ligands within the same cell (cis interaction) is well established, and its in vivo relevance to the lateral inhibition circuit underlying neuronal differentiation in the fly eye has been demonstrated (8). However, its in vivo relevance for SOP differentiation has been controversial. Moreover, its possible contribution to the reliable information processing of the lateral inhibition circuit has not previously been examined. Our study proposes a rationale for its use as part of the lateral inhibition circuit.

An essential prediction of our model is that, at least in the initial stage of SOP selection, the interaction in cis is more potent than the trans interaction. The in vivo efficiency of cis versus trans interactions has mainly been tested by overexpressing Delta or Serrate in a clonal fashion within the developing wing (25, 32, 33). In these experiments, Notch activity was observed in cells surrounding the clones, but not within the clone. These results suggest an efficient cis interaction because Notch in every cell is subject to similar amounts of ligand in cis and in trans. Moreover, the affinity of the interaction in trans is regulated by the E3 ubiquitin ligase, Neuralized (25). Thus, the relative potency of the cis versus the trans interaction may vary during the process of SOP selection. Typically, Neuralized is detected only in the selected SOP (23), supporting the model assessment that it is the cis, rather than the trans, interaction that dominates during the initial stages of SOP selection. Further biochemical experiments, however, are required to more rigorously examine this prediction.

In conclusion, we suggest that lateral inhibition circuits are prone to errors because of the inevitable delay in executing inhibition and because of the occasional instances in which variability between the cells is small. Minimizing signaling delay is thus critical for efficient use of the lateral inhibition mechanism, and we propose that cis interaction plays a pivotal role in minimizing the effective delay, enabling robust selection. Thus, the need to maintain a robust function poses strict restriction on the design of lateral inhibition circuits.

Materials and Methods

Single-component model of a lateral inhibition process

In the one-component model, the lateral inhibition circuit in each cell i is simplified to include a single component only, Mi. The dynamics of M is defined by the following Langevin equation:dMidt=βf(Mi,Mj,t)λMi+η(t)(M.1)

with f(Mi,Mj,t) given byf(Mi,Mj,t)=(1θ(krMi(t))θ(Mj(t)kin))(M.2)

Here, λ denotes the degradation rate, β the maximal production rate of M, and θ the Heaviside step function [θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0]. Lateral inhibition is initiated for Mj > kin, which inhibits Mi production unless Mi has passed the threshold for refractoriness, Mi > kr. For simplicity, we assume that the onset of lateral inhibition, as well as refractoriness, is step-like. Finally, a cell i is considered to be selected as an SOP when Mi passes the threshold for irreversible selection, Mi > ks.

Noise is introduced by the η(t) term, which is a random variable chosen from a normal distribution with η(t)=0;η(t)η(t)=(βf(Mi,Mj,t)+λM)δ(tt)(M.3)

All simulations were performed with Matlab; we iteratively followed the stochastic accumulation of Mi in a time step of dt = 10 s. In each time step, the amount of production and degradation was taken from Poisson distribution with mean βf(Mi,Mj,t)dt and λMidt, respectively.

All simulations were performed with the following parameters:β=0.2moleculemin,λ=1200min1

Additional parameters for the simulations reported in Fig. 2B are ks = 30 molecules and kin = kr = 20 molecules.

A time delay was introduced by substituting into Eqs. M.1 and M.2 Mj level at earlier time, t′, rather than its value at time t (where t is the time when the derivative dMi/dt is evaluated).

To model the system in the presence of an inhibition delay, τin, we have replaced Eq. M.2 with f(Mi,Mj,t)=(1θ(krMi(t))θ(Mj(tτin)kin))(M.4)

To capture a scenario that also includes a delayed relief of the repression (defined as τcycle), we define a more general form of f(Mi,Mj,t):f(Mi,Mj,t)=(1θ(krMi(t))θ(max(Mj(t))kin))(M.5)

where the maximum is taken over Mj level at the following time range, t’:t:tτinτcycletτin(M.6)

Specific parameters for the simulations reported in Fig. 2D are ks = 30 molecules, kin = 20 molecules, and τin = τs = 10 min.

The different scenarios are obtained with the following parameters:

  1. (i) Iteration: kr = 30 molecules, τcycle = 0 min

  2. (ii) No relief: kr = 30 molecules, τcycle→∞

  3. (iii) Refraction: kr = 20 molecules, τcycle = 0 min

For the simulations reported in Fig. 3A, we set f(Pi,Pj,t) = 1.

Fly strains

The following alleles were used: N55e11, FRT101 (Couso and Martinez Arias, 1994); DlRevF10, FRT82B (Thomas and Knust, 1991); SerRX82, FRT82B (Haenlin and Campos-Ortega, 1990); DlRevF10, SerRX82, FRT82B (Thomas and Knust, 1991; Haenlin and Campos-Ortega, 1990); E(spl)mα-DsRed.T4-NLS (Barolo and Posakony, 2004); and y.w, OregonR, CantonS (Bloomington).

Live imaging of SOP selection in pupal discs

Intact pupae were collected at ~8 to 10 hours after puparium formation (APF), submerged in halocarbon oil, and subjected to time-lapsed fluorescent microscopy at 2-min intervals under a Zeiss LSM710 system. Time-lapsed photography of scutum small bristle (microchaetes) SOP selection was performed until pupal head eversion (~12 hours APF), a process that causes movement of the monitored cells and their detachment from the pupal cuticle. Confocal images were z-stacked in 5-μm steps to form a vertical axis of ~100 μm. Image analysis (Fig. 2, E to F) was performed with a Zeiss LSM image browser, selecting the maximal intensity pixel from a set of z-axis positions.

Acknowledgments

Acknowledgments: We thank J. W. Posakony, H. J. Bellen, F. B. Gao, E. C. Lai, F. Schweisguth, D. Van Meyel, M. Milan, B. Z. Shilo, and the Bloomington Stock Center for fly stocks and B. Nadler, B. Z. Shilo, and all our laboratory members for helpful discussions. Funding: O.B. is supported by the Azrieli Foundation and the Kahn Family Research Center for Systems Biology of the Human Cell. E.H. is the incumbent of the Helen and Milton A. Kimmelman Career Development Chair. This work was supported by the Israel Science Founction (E.H. and N.B.), the European Research Council (IDEA), Minerva, and the Hellen and Martin Kimmel award for innovative investigations (N.B.). Author contributions: O.B., E.H., and N.B. conceived the research. O.B. performed all the computational analysis and experiments. D.R. assisted with the experiments. O.B., E.H., and N.B. wrote the paper. Competing interests: The authors declare no competing interests.

Supplementary Materials

www.sciencesignaling.org/cgi/content/full/3/129/ra51/DC1

Section S1. Probabilistic model of lateral inhibition: rules for successful selection.

Section S2. Error rate-formal calculation.

Section S3. Error rate in a purely stochastic lateral inhibition process.

Section S4. Error rate in a lateral inhibition process with a pre-pattern bias.

Section S5. Error rate in an iterative process.

Section S6. Transcription-based feedback model of lateral inhibition process: the effective time delay, τeff.

Section S7. Cis inhibition model of lateral inhibition process: the effective time delay, τeff.

Section S8. Predicted error rates for different combinations of Notch, Delta, and Serrate heterozygotes.

Section S9. A smooth inhibition function leads to a higher error rate.

Section S10. Validation for the use of the mα reporter.

Fig. S1. Minimizing inhibition delay reduces error rate in a lateral inhibition process with a pre-pattern bias.

Fig. S2. Reduction of error rate in an iterative process.

Fig. S3. Simulation results showing the error rates as a function of the Hill coefficient.

Fig. S4. Validation of SOP detection using the mα reporter.

Table S1. Summary of the specific genotypes tested for ectopic selection of anterior scutellar (aSC) bristles.

References

References and Notes

  1. This assumption of a threshold-like inhibition is not limiting and provides in fact a lower bound for the error rate; see section S9 and fig. S3.
  2. Note that this formalism is also equivalent to a system that displays a ‘no-relief,’ rather than refractory dynamics.
  3. The recombinant allele of Delta-Serrate used to generate the triple heterozygote displayed an error rate that was significantly lower than that observed generating the double heterozygotes Delta-Serrate from different backgrounds. In all cases, error rates were measured after introducing the heterozygote allele to within the same wild-type background (yw) (table S1).
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