While most processes in biology are highly deterministic, stochastic mechanisms are sometimes used to increase cellular diversity. In human and Drosophila eyes, photoreceptors sensitive to different wavelengths of light are distributed in stochastic patterns, and one such patterning system has been analyzed in detail in the Drosophila retina. Interestingly, some species in the dipteran family Dolichopodidae (the “long legged” flies, or “Doli”) instead exhibit highly orderly deterministic eye patterns. In these species, alternating columns of ommatidia (unit eyes) produce corneal lenses of different colors. Occasional perturbations in some individuals disrupt the regular columns in a way that suggests that patterning occurs via a posterior-to-anterior signaling relay during development, and that specification follows a local, cellular-automaton-like rule. We hypothesize that the regulatory mechanisms that pattern the eye are largely conserved among flies and that the difference between unordered Drosophila and ordered dolichopodid eyes can be explained in terms of relative strengths of signaling interactions rather than a rewiring of the regulatory network itself. We present a simple stochastic model that is capable of explaining both the stochastic Drosophila eye and the striped pattern of Dolichopodidae eyes and thereby characterize the least number of underlying developmental rules necessary to produce both stochastic and deterministic patterns. We show that only small changes to model parameters are needed to also reproduce intermediate, semi-random patterns observed in another Doli species, and quantification of ommatidial distributions in these eyes suggests that their patterning follows similar rules.
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 a particular probability 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 Drosophila eye [5,6], a complex organ whose development has been the subject of great scrutiny [7,8]. Our interest focuses on the patterning of photoreceptors (PRs) that are involved in color vision: two types of ommatidia express different combinations of color-sensitive photopigments Rhodopsins in their “inner” R7 and R8 PRs, and these two ommatidial types are randomly distributed across the retina [9]. This can be visualized via staining with antibodies against the green-sensitive (Rh6) or blue-sensitive (Rh5) Rhodopsins expressed in R8 PR cells (Fig 1a). A similar stochastic pattern exists in R7 PR cells for two UV Rhodopsins, Rh3 and Rh4 [10]. Stochastic on or off expression of the transcription factor Spineless in the R7 PRs 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) that produce two distinct corneal lens colors (Fig 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 may share the same underlying regulatory mechanisms with similar downstream effectors, but which differ in how the expression of few critical upstream regulators is controlled. Thus, it might be possible to predict the rules that underlie the control of stochastic vs. deterministic patterning, and might underlie the evolutionary conversion 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 three fly species. Our model is also predictive and applicable to patterns observed in the eyes of other flies; as an example, we present predictions for the eyes of another species of Dolichopodidae that displays intermediate patterns. 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 signaling 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 color 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 color 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 PRs [8]. Two types of so-called pale and yellow ommatidia, are randomly distributed across the retina in a 35: 65 ratio [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] (Fig 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 three fly species (stochastic, ordered and semi-ordered) to a single switching mechanism.
We assume that the choice of a color 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, the two alternate colors, green and red, are denoted by 1 and 0 respectively. In the hexagonal lattice formed by the ommatidia, the color choice of the ommatidium in the ith row and jth column is defined by the element aij (which is either 1 or 0) in an n × m matrix. Two competing effects determine the value of a given element: The first is a default probability of being in the state 1 (green) for every element in a vertical column. The second is that of the subtype correlation between the ommatidium and its nearest neighbors in the preceding column. The default probability pj (S|X) of all sites in a column j to be green, can be expressed as: p j (S | X) = { P 0 X j ≤ X 0 1 - P 0 X j > X 0 (1) with P0 being a constant. This expresses the fact that, for all values smaller than a threshold X0, the probability for the column to be green is given by P0: this changes to 1 − P0 when the threshold is exceeded. High probabilities of being green, as mentioned above, are related to the expression of Spineless. The expression of S leads in its turn to a rapid change in the value of X over column j + 1, which can be simply modelled by the following linear equation: X j + 1 = γ - β p j (S | X) (2) where γ and β are constants. If β is negative, successive columns are anticorrelated, in order to model the case of Doli eyes (Fig 1a), while the positive sign is appropriate for a hypothetical fly that would have a fully homogeneous retina. On the other hand, the constant γ, which parametrizes the sensitivity of the switch, is strictly positive. In order to build an extremely ordered alternating pattern such as that of Doli shown in Fig 1b, we can choose the constant P0 to be very small which guarantees almost uniform color in each column. We illustrate the case for P0 being very small in the following: In Fig 2a, P0 = 0. 0001, so that if, at the jth column, Xj is greater than the threshold, then the default probability pj of the column to be green is very close to 1 (1 − P0 = 0. 9999, from Eq 1). With suitably chosen constants (e. g. β = 8, γ = 10, X0 = 5 as chosen here), Eq 2 implies that the value of X at column j + 1, Xj+1 ∼ 2, which is less than the threshold X0. In turn, the first line of Eq 1 yields pj+1 = 0. 0001, i. e. column j + 1 is red with very high probability. The ordered stripes of red and green are thus built across the eye as depicted in Fig 2a. For the ordered retina, the negative sign of β means that an ordered initial column with an above-threshold value of X, will, from Eq 2, give rise to a constant value of X in all successive columns (γ − β (1 − P0) ), so that a column that is initially green will stay green forever, that is, all subsequent columns become the same color as the very first one. Some Doli species, such as this example from the Chrysosoma genus (Fig 1d), have partially ordered eyes that are intermediate between Condylostylus and Drosophila retinas. These patterns can be modelled as perturbations of the uniform fly, as seen in Fig 2d. Fig 3 is an illustration of the quantitative behavior of Eqs 1 and 2 as a function of all the parameters β, γ, P0 and X0, with γ and X0 scaled by β for convenience. Fixing the scaled value X0/β at a sample value of 0. 2, we obtain the regions of phase space dominated by Doli-type or uniform retinal pattern. The blue and purple regions correspond to uniform retinas, the green to the Doli retina, while the white region corresponds to physically inadmissible values where X0 (as a biological factor) takes negative values. The overlap of the blue, green and purple region for P0 > 0. 5 indicates that the green region or the oscillating Xj will not occur anymore. The value of Xj will be fixed at either γ − β (1 − P0) or γ − βP0 in the overlapped region depending on its initial value. At the other end of the spectrum from Doli, the Drosophila retina contains ommatidia where R8 cells express either green- or blue-sensitive Rhodopsins that are randomly distributed, with a bias for green-sensitive (yellow, 65%) vs. blue-sensitive (pale, 35%). This pattern is fully random [5,22–24]. This implies that the choice in every ommatidium is fully independent of all its neighbors, indicating that there is a total absence of correlations and full stochasticity: no auxiliary equation like Eq 2 is therefore needed. On the other hand, the bias can be accommodated by choosing the default probabilities of the two subtypes to be unequal. The analog of Eq 3 in this case becomes P i j D r o s o (S, l) = p j D r o s o (S | X) (4) where Pij is the probability that the ommatidium ij is green. Fig 2b shows a sample configuration simulated with P0 = 0. 35 (see Eq 1), to be compared with an image of the Drosophila eye (Fig 1a). Retinal patterns in other fly species may be conceived as a “mixture” of characteristics of the disorder in Drosophila and Doli or a fully ordered retina, respectively. The simplest way to achieve this from a mathematical perspective is to consider linear combinations of the transition probabilities of the form P j α (S | X) = α p j O r d e r e d F l y (S | X) + (1 - α) p j D r o s o (S | X), (5) and P j α (S | X) = α p j D o l i (S | X) + (1 - α) p j D r o s o (S | X), (6) respectively. The parameter α ∈ [0,1] quantifies the relative influence of the stochastic, Drosophila-like mechanism. Eq 5 applies for β < 0, and Eq 6 for β > 0. We use this formalism to analyze the eye of a partially ordered fly from the Chrysosoma genus (Fig 1d) where, as for Doli eyes, the pattern is that of corneal colors rather than Rhodopsin expressed in photoreceptors, which are not known in this species. Mathematically, this is a finite α case with β negative, corresponding to the first of Eq 5. Our simulations (Fig 2d) agree qualitatively with the image of a real fly shown in Fig 1d; however, a detailed experimental study of its geometrical correlations needs to be performed to get more quantitative agreement. This formalism allows us to present a generic phase diagram (Fig 4), in which any given fly eye can be defined; the specific location, along with the theory above, enables us to identify the specific mechanisms associated with the morphogenesis, e. g. whether the fly is derived from a perturbation of the Uniform Fly, the hypothetical fly with completely ordered eye colors, or from Doli, and so on. This phase diagram is calculated based on the horizontal and vertical autocorrelations detailed below (see Materials and Methods). Fig 4G corresponds to the case when one of two colors (in this case green) is predominant. The horizontal axis shows the value of α that varies from 0 to 1 symmetrically on each side of the figure; the right side corresponds to positive β, and the left side to negative β. The vertical axis gives the value of P 0 D r o s o. Panels A-I show sample configurations. Configuration A is pure Doli-like, where β is positive and α = 1 (the two arrows indicate that the value of P 0 D r o s o is irrelevant, see Eq 6). Configuration B represents a slight deviation from Doli, with β > 0, α = 0. 9 and P0 = 0. 4. In configuration C, the reduced value of α = 0. 65 leads to more and more deviations from Doli; here, P0 = 0. 9. Configuration D corresponds to parameters α = 0. 65 and P0 = 0. 1; this is a near mirror image of C, because of the flipped value of P0 with respect to C. The β < 0 region in the negative x-axis is largely disordered. The probability of the so-called Uniform Fly is considered as p j O r d e r e d F l y (S | X) = 0. 99 so that, e. g. the extreme configuration which occurs at α = 1 is 99 percent green. Configurations E and F are drawn based on identical values of α close to zero but for two opposite points on the vertical axis corresponding to P 0 D r o s o = 0. 1 and P 0 D r o s o = 0. 9. Therefore, a Drosophila-dominant behavior with different percentages of greens and reds is observed in these panels. As α increases to the left, the effect of the Drosophila term in the first of Eq 6 decreases, so that configurations tend to become more and more a homogeneous green as will be seen in configurations G, I and H which correspond to α = 0. 9 and P0 = 0. 9,0. 55 and 0. 1 respectively. We next discuss the significance of our results and seek to relate the patterns we predict theoretically to simulated as well as real data. Since the stochasticity inherent in the model suggests that almost all parameter combinations generate almost all patterns with non-zero probability, the pertinent question to ask is: which parameter combination most likely generates a given pattern? The aim of this exercise is twofold. First, we show using data simulated by our model, to what extent it is possible to infer the model parameters consistently from the patterns generated by the same model. This provides a baseline for the analysis of real-life images where we strive to infer the most likely parameters for a given image of the fly eye. These parameters then provide the mostly likely explanation for the mechanism that has generated the observed pattern via the phase diagram (Fig 4) given above. A few technical considerations are in order: The parameters β and γ, cf. Eq (2), themselves are not well-suited for inference because large fractions of the (β, γ) parameter space generate the same behaviour (see Fig 4). We replace β and γ by a discrete variable m with only 4 distinct values m ∈ {fr, fg, ar, ag}. These represent the combination of two binary variables (i) dynamic mode: fixed point / alternation (ii) initial condition: green probability high / low. Furthermore, we discretize the values of the parameter α to be multiples of 0. 01, i. e. α ∈ V (α) ≔ {0. 0,0. 0. 1,0. 02, …, 1. 00}. We restrict ϵ ∈ V (ϵ) ≔ {0. 0,0. 1,0. 2,0. 3,0. 4,0. 5} and P0 ∈ V (P0) ≔ {0. 0,0. 1}. This discretization of the inference problem makes it computationally easier. It is justified by the fact that small variations in P0 affect probabilities in a way similar to small variations in α. In particular the parameters α and P0 are fully interdependent for a fixed point behavior (m ∈ {fr, fg}). In this case, we set P0 = 0. We keep the spatial range parameter at k = 1. 0 throughout, thus not subjecting it to inference. We first apply parameter inference to simulated patterns generated by the model itself in order verify that parameters can indeed be inferred consistently. For each realization, we (i) draw parameter values uniformly at random from the set of eligible combinations defined in the preceding paragraph; (ii) generate a pattern by the model with these parameter values, which we call the true values; (iii) find an eligible combination of parameter values that maximizes the probability of the model generating the pattern from step (ii) (maximum likelihood). Fig 5 shows the results for 10000 independent realizations of the inference using patterns of 50 columns and 30 rows. The mixing parameter α is inferred with good precision over the entire range. Values of the other parameters are also inferred with large precision for sufficiently large α. For small α, the color assignment is dominated by the fixed probabilities of Drosophila, so that the parameters m, ϵ and P0 have less influence on the pattern. Inference is expected to be more difficult for smaller values of α. As an application of the model to real data, we perform similar parameter inference on photoreceptor distribution data collected from the eyes of individuals in the “intermediate” Chrysosoma species. See Fig. S1 in S1 Text for details on images and processing of data. Parameters are discretized for the inference; α is in the set {0. 01,0. 02, …, 0. 99}; P0 ∈ {0. 01,0. 02,0. 04,0. 08,0. 16}; ϵ ∈ {0,0. 001,0. 002, …, 0. 010}; and k ∈ {1. 0,4. 0,9. 0,16. 0,32. 0, …, 625. 0}. The results of maximum likelihood parameter inference are as follows: All in all, the basic behaviour (red fixed point) is found consistently in all eyes. In the other parameters, we have strong fluctuations between samples; in some cases (large ϵ and large α), the stochasticity is described by the large probability of error propagation; in the case of small α, the patterning resembles Drosophila. In future studies, stronger support for the model could be obtained by analyzing data from additional species. Ideally, the species could be distinguished by how their parameters fall into disjoint regions.
Statistical physics can be usefully applied to biology when searching for organizational principles [25–27]. The present study is such an example: we present a minimal model that describes patterns observed in the retinas of Drosophila and Dolichopodidae flies. Via a generic phase diagram, we are able to predict parameters that might underlie a range of patterns found on the eyes of various fly species. Conversely, model parameters can be estimated from images of retinal patterns. The essential ingredients of the model are stochasticity and correlations: an ommatidium chooses its color state depending on the competition between its gene-expression-induced probability to be a specific color (e. g. the expression of Spineless) and the influence of the ommatidium in the preceding column along the progression path of the morphogenetic furrow. Intercolumnar correlations greatly outweigh the effect of stochasticity in Doli, while the reverse is the case for Drosophila. This formalism also allows us to probe how mistakes propagate in the retina, for example in Doli. For instance, supposing an ommatidium in a column is a “mistake”, i. e. is of the wrong color, our formalism suggests that two scenarios are possible. If the mistake is isolated, it will soon be resolved as the morphogenetic furrow progresses. However, a cluster of mistakes can lead to interesting domain formation, with interfacial roughness as in, say, crystalline systems where dislocations define the area of interfacial disorder [28]. This is in accord with experimental observations [5,20]. The richness of our formalism, however, goes well beyond the description of known species to the prediction of those that are as yet undiscovered. In terms of simple model parameters, we are able to generate a global phase diagram for flies with binary color choices and allow for important clues to the morphogenetic mechanisms at play. The suggested parameter values for the Chrysosoma species are an example of this power, although we emphasize that in the absence of quantitative experimental data about the molecules involved and variables such as their diffusion constants, we have here aimed at only qualitative agreement. We hope this work will motivate detailed quantitative analysis of experimentally observed patterns, as well as genetic analysis of factors involved in the expression of color in fly eyes, so that our predictions can be put to the test. Once our model is adequately tested for two-color retinae, we will be able to extend our analysis to flies and other insects with more complex color patterning.
A simple model is able to account for a diversity of photoreceptor patterns in different fly species, ranging from highly deterministic to fully random.
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Patterning the insect eye: From stochastic to deterministic mechanisms