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. 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.

with the step function Assuming a stripe pattern, the Θ-term in Eq. (1) gives the a-value expected to be dominant in column j. For p j > p j+1 , the probability of green ommatidia (a i,j = 1) in column j is larger than that in the adjacent column j. Green ommatidia are the standard in this column j, and a red ommatidium a i,j = 0 in a row i is considered an error. Likewise, green ommatidia are considered errors in red-dominated columns (p j < p j+1 ), where preceding the description holds analogously by exchanging 0 and 1 entries of a. An error in column j potentially perturbs the nearby ommatidia in the following column j + 1. The influence is assumed to decay exponentially with the distance between ommatidia; hence the exponential factor with a length scale parameter k in Eq. (1). Influences from all errors in the previous column sum up linearly (summation with row index h) and are scaled with a coupling strength .
Accounting for the hexagonal lattice, we distinguish two index sets for the row index. The set is the usual set of integer index values, used for the columns j with j even. In columns with odd j, sites have row indices in

Analysis of correlations in patterns.
Given a pattern (a ij ) with row index i ∈ {1, . . . , n} and column index j ∈ {1, . . . , m}, the horizontal autocorrelation coefficient is defined as with the pattern's mean value and variance Likewise, the vertical autocorrelation coefficient is defined as Since we are studying stochastic pattern generation, r realizations (r 1) are performed for a given set of parameter values (α, β, P 0 ). For the pattern generated in each realization k (1 ≤ k ≤ r), the autocorrelation coefficients R h k and R v k are computed. Then these coefficients are averaged Image and data processing Images of thirty retinas from fifteen individuals (both left and right eyes) were collected using a Leica M80 stereomicroscope and IC80 HD camera. Positions of (x, y) of individual ommatidia were identified using a difference of gaussian blob detection approach as used to classify cell nuclei by Bothma et al. [1]. For each eye, the spatial coordinates of ommatida were scaled so the nearest neighbor of an ommatidium is at distance 1.0 on average. The ratio of red vs. green intensity was used to classify ommatidial type. This way we obtained, for each eye, a list of R ommatidia, The i-th ommatidium is characterized by its spatial coordinates x i and y i and its type t i ∈ {red, green}. Fig. S1 shows images of two example of eyes from the semi-ordered Chrysosoma species; one is more ordered than the other (high vs. low values for alpha). Real images of eyes are shown on the left, with examples of rotated and zoomed "classified" images on the right. This classified red/green set was used to generate coordinates for each unit eye and assign unit eye type.
The assignment of a column index c i to each ommatidium i is done as follows.
2. The ommatidium closest to the center of mass of all ommatidia is assigned column 0.