Patterning the insect eye: from stochastic to deterministic mechanisms

While most processes in biology are highly deterministic, stochastic mechanisms are sometimes used to increase cellular diversity, such as in the specification of sensory receptors. In the human and Drosophila eye, photoreceptors sensitive to various wavelengths of light are distributed randomly across the retina. Mechanisms that underlie stochastic cell fate specification have been analysed in detail in the Drosophila retina. In contrast, the retinas of another group of dipteran flies exhibit highly ordered patterns. Species in the Dolichopodidae, the “long-legged” flies, have regular alternating columns of two types of ommatidia (unit eyes), each producing corneal lenses of different colours. Individual flies sometimes exhibit perturbations of this orderly pattern, with “mistakes” producing changes in pattern that can propagate across the entire eye, suggesting that the underlying developmental mechanisms follow local, cellular-automaton-like rules. We hypothesize that the regulatory circuitry patterning the eye is largely conserved among flies such that the difference between the Drosophila and Dolichopodidae eyes should be explicable in terms of relative interaction strengths, rather than requiring a rewiring of the regulatory network. We present a simple stochastic model which, among its other predictions, is capable of explaining both the random Drosophila eye and the ordered, striped pattern of Dolichopodidae. INTRODUCTION The development of multicellular animals is highly reproducible, with deterministic and orderly processes generating reliable outcomes. Segment boundaries form in the proper place and cell types are set aside in specific proportions in differentiating tissues. Underlying these seemingly precise developmental outcomes, though, are inherently stochastic transcriptional events, ​e.g. ​decisions to express or not express key regulators of cell fate (1, 2). Varying amounts of activating or repressive input can bias these decisions strongly one way or the other, producing seemingly deterministic on or off outcomes, resulting in distinct boundaries and specific spatial patterns (3, 4). The distribution of these inputs depends largely on lineage and positional information within an embryo or tissue. In another class of cell fate decisions, stochastic cell-intrinsic mechanisms instead produce particular probabilities of taking one fate or another in otherwise equivalent cells (5). In their own way, these stochastic decisions are highly regulated to take place in specific tissue types and to produce reliable proportions of one cell fate ​vs.​ another. How such probabilistic patterning mechanisms might be switched between stochastic and deterministic is a question to which the tools of statistical physics can meaningfully be applied. An example of stochastic patterning occurs in the fly eye (5, 6), a complex organ whose development has been the subject of great scrutiny (7, 8). Our interest focuses on the patterning of two photoreceptors (PRs) that are involved in colour vision; the two “inner PRs” R7 and R8 are randomly distributed across the retina (9), as seen via staining with antibodies against the green-sensitive or blue-sensitive photopigments, the Rhodopsins in R8 cells (Figure 1a). A similar stochastic pattern exists in R7 cells for two UV Rhodopsins, Rh3 and Rh4 (10). Stochastic on or off expression of the transcription factor Spineless in the R7 PR controls ommatidial type, and therefore the overall random pattern (5). In contrast, another group of flies in the family Dolichopodidae (referred to here as “Doli”) have ordered retinal patterning with alternating columns of ommatidia (the individual units of the adult compound eye) which produce two distinct corneal lens colours (Figure 1b). The patterning mechanisms that underlie both differentiation of PR types (e.g. R7 vs. R8) and stochastic patterning across ommatidia have been shown to be largely conserved between ​Drosophila and butterflies (11, 12). Considering the apparent similarities between the ​Drosophila and Doli eye, it is tempting to suggest that the cell fate decisions involved in stochastic vs. non-stochastic patterning share the same underlying regulatory mechanisms with similar downstream effectors, but differ in a few critical upstream steps, and it might be possible to establish the rules that governed evolution from one mode of patterning to the other. In this work, we present a simple mathematical model for such a regulatory mechanism, and compare our results with experimental data from the two fly species. Our model is also predictive and applicable to patterns observed in the eyes of other flies; we present predictions for the eyes of another species of Dolichopodidae that displays intermediate patterns as an example. The adult eye is a geometrically regular structure composed of hexagonal unit eyes packed into a grid. Patterning begins with the progression of the morphogenetic furrow, a posterior-to-anterior wave of differentiation. Sequential rounds of signalling produce ~25 highly ordered rows of ​∼ 30 ommatidia each to make up a total of 800 ommatidia per eye (13). Each ommatidium contains eight PRs and accessory cells: the six “outer PRs” (R1-R6) express a broad-spectrum Rhodopsin, Rh1, and are required for motion and dim light vision. The two “inner PRs” (R7 and R8) each express different Rhodopsins and are used for colour discrimination and polarized light vision (8, 14–16). A detailed mathematical model for much of the process of eye formation has recently been formulated (17). However, that model does not address the stochastic distribution of colour photoreceptors, which is the subject of this paper. There are three main ommatidial subtypes in Drosophila, which are defined by the combination of Rhodopsin photopigments expressed in their R7 and R8 photoreceptors. Two of these, the so-called ‘pale’ and ‘yellow’ ommatidia, are randomly distributed across the retina in the ratio 35:65 (5, 9). The pale ommatidia express UV-sensitive Rh3 in R7 and blue-sensitive Rh5 in R8 and are used for the discrimination of short-wavelength light (5, 8). The yellow ommatidia express longer UV-sensitive Rh4 in R7 and green-sensitive Rh6 in R8 and are used for the discrimination of longer wavelengths (18) (Figure 1a). A third subtype found in the dorsal rim area (DRA) is used for the detection of the vector of light polarization (19). The stochastic distribution of yellow and pale ommatidia in Drosophila is controlled at a single upstream node in the retinal regulatory network by the stochastic expression of the transcription factor Spineless (Ss) in a subset of R7 cells (5). In Doli, where the patterning is highly ordered, Ss might also be responsible for Rhodopsin expression as the eyes appear to develop in highly similar ways; it also seems likely that many of the interactions in the eye regulatory network are conserved between Doli and Drosophila (11, 12). The generation of the very different patterns observed might thus be due to changes in the initial expression of Ss. In this paper, we present a simple mathematical model that captures the essence of these ideas, by attributing the diverse patterning in the two fly species to a single switching mechanism. Model of retinal patterning. Initial eye development proceeds via a complex set of interactions between cell signalling and changes in target gene expression as new cell types are sequentially recruited (13). After the progression of the morphogenetic furrow and recruitment of all cell types that will make up the adult retina, cell biological processes begin to shape and structure the ommatidia. The PRs produce microvillae that make up the rhabdomeres, the light gathering structures. The decision to express Ss (or not) determines the choice of Rhodopsins in the inner photoreceptors R7 and R8, and consequently, the colour sensitivity of the ommatidium. In Drosophila, this decision leads to a random distribution of Rhodopsins in pale or yellow ommatidia (5). In contrast, Doli eyes instead show an orderly pattern of alternating columns, such as observed in the genus ​Condylostylus (Fig. 1b). Interestingly, we observed occasional perturbations in patterning in wild-collected Doli (Fig.1c). In some individuals, when multiple errors occur in adjacent or nearly adjacent ommatidia, errors in patterning sometimes propagate in an anterior direction from the initial column containing mistakes (see Fig. 1c for an example). In some animals many errors occurred in approximately the same column on the anterior-posterior axis in both retinas, suggesting a developmental cause such as thermal stress during a specific time during the migration of the morphogenetic furrow. Whatever the cause, the subsequent propagation of errors in the direction of the morphogenetic furrow suggests that initially local, cellular-automaton-like rules are at work. With the idea that local signalling might set up more global patterns of alternating columns in mind, we made a key assumption: the regulatory circuitry being largely conserved among flies, any differences between fly species might be explicable in terms of relative strengths of interactions, rather than through an entire rewiring of the regulatory network itself. In this spirit, we assume the existence of a single gene product X that is required to activate the switch referred to above; thus, a (deterministic or stochastic) decision to express a specific Rhodopsin can be triggered only in the presence of ​X​. We also assume that the coupling strength of the factor X with the switching mechanism which eventually determines the eye colour, differs between fly species and can be influenced by changes in the unknown factor X and/or in the switching mechanism itself. Furthermore, the expression of ​X in an ommatidium should itself depend on the decisions made in the neighbouring ommatidium that developed just before it during the progression of the morphogenetic furrow; accordingly, in our model, we choose ​X at a given ommatidium to be the (weighted) average of its values in its preceding nearest neighbours. To sum up: ​X is a factor that diffuses with the furrow, whose values at a given ommatidium are correlated with those of its previous neighbours, and which is responsible for making local decisions at the point where photoreceptors acquire their identity. These different values of ​X may, depending on their magnitudes, be able to activate the switch required to express a specific Rhodopsin. For the two cases of immediate interest, this switch is Ss whose expression levels determine whether ommatidia are yellow or pale in Drosophila, and perhaps whether stripes are green or red in Doli. However, for the general case, we define a conditional probability which embodies the dependence of the expression of an arbitrary factor S in a given ommatidium on the expression level of the diffusing factor ​X introduced above. This includes the limiting case when (as in Drosophila), the colour decision becomes independent of ​X​; here, the equation applies. Dynamically, pattern formation in the fly eye proceeds column by column: that is, spatial organisation and colour choices are made for every column of future ommatidia traversed by the morphogenetic furrow in its progression across the eye (8, 20). This implies a discrete temporal separation of the columns of ommatidia, which makes their colour decision the final step in their formation. The specific parameters clearly depend on the fly species. The Drosophila eye has random ordering with a bias for the yellow ommatidial type. Dolis instead have ordered retinal patterning, where columns of ommatidia with alternating corneal lens colours are found. This suggests that the colour decision is made by a bistable switch. Some clues about the nature of this switch can be found in Doli retinas that contain mistakes, where the regular ordering of red and green columns is sometimes interrupted. These errors can be propagated locally, sometimes over multiple subsequent columns, which suggests that the process that produces order is itself based on biasing an inherently stochastic decision toward a reliable outcome, but which can be interrupted by external conditions. For example, local propagation of errors makes it unlikely that the entire field is patterned using global retina-wide gradients of positional information. It is possible that perturbations during development might increase the frequency of mistakes during patterning near the morphogenetic furrow at the time of the perturbation, as suggested by the presence of multiple errors per column and in the same column in both eyes. These phenomena suggest that colour decisions in a column can have a local influence on their neighbours. Taken together, these features suggest a unified theory that contains two ingredients: one, a stochastic choice element, and two, a correlation between the expression of colour choices in adjacent columns. In the case of Drosophila, the second of these can be set to zero to reflect the purely random spatial ordering of colours. Accordingly, in the following section, we first define our parameters in the context of the Doli eye and look at its dynamic formation, discussing the propagation of perturbations. Next, we describe the formation of the Drosophila eye in the context of our model. Patterning of the Doli eye. We assume that the choice of a colour in each column is essentially complete by the time the morphogenetic furrow moves on to the next column. Thus, we regard fate establishment as instantaneous on the timescale at which the full ordering process occurs in the next column. Consistent with this assumption, Ss expression in Drosophila starts almost immediately after the morphogenetic furrow (21). In Doli, we denote the two alternate colours, green and red, by 1 and 0 respectively. In our formalism, the colour choice of the ommatidium in the ​i​th row and ​j​th column of the embedding hexagonal close-packed lattice is defined by the element (which is either 1 or 0) in an ​n x m matrix. Two competing effects determine the value of a given element: The first is a default probability of being in state 1 (green) for every element in a vertical column. The second is that of the subtype correlation between the ommatidium and its nearest neighbours in the preceding column. Default probabilities. In our model for Doli, the expression of Ss leads to ommatidia being green. We are thus led to define the default probability of all sites in a column j to be green as: