Fig 1.
(a) Stochastic distribution of UV-sensitive Rhodopsin genes (Rh3 and Rh4) in R7 cells of the Drosophila retina. (b) Striped pattern of red alternating with yellow/green ommatidia in the retina in a Condylostylus species fly, family Dolichopodidae. Single “error” circled in grey. (c) Image of another Condylostylus individual eye with several errors. Anterior is to the left. In some cases, especially where several errors occur in proximity, the fate of the next most anterior column (to the left) is also modified. Such errors propagate across the eye from posterior to anterior. Note that isolated errors (marked with grey circles) do not propagate. (d) Retina of a partially ordered Doli species in the Chrysosoma genus.
Fig 2.
Simulation results for a model fly eye of size 30 × 50 obtained via the general response function of our model (Eq 6).
The parameter α quantifies the relative importance of stochastic decisions. In: (a) α = 1. This models a perfect Doli fly, where there are no mistakes; (b) α = 0. This models Drosophila, where the two colors are randomly distributed, with a bias towards the green; (c) α = 1. This models a Doli fly a perturbed column (eighth from right). Notice that the perturbations propagate leftward to the end, giving rise to a domain with a discernible boundary. (d) α = 0.7. The parameters ϵ (local speckle correlation coefficient) = 0.95, and
were chosen with a view to match, at least qualitatively, the patterning of the intermediate, partially organized Chrysosoma sp. shown in Fig 1d.
Fig 3.
Plot of the behaviour of Xj based on Eq 2 as a function of the key parameters in the problem with the (scaled) threshold value of X0 shown by the dotted line.
The purple and blue regions correspond to the parameters which lead to fixed values Xj = γ − β(1 − P0) and Xj = γ − βP0 respectively. Such fixed values become stable states of Xj, corresponding therefore to the mythical “Uniform Fly”. The green region represents the range of parameter values where Xj oscillates between the above fixed points, corresponding therefore to Doli. The white region corresponds to physically inadmissible values where X0 (as a biological factor) takes negative values. In the region where P0 > 0.5, blue, green and purple region overlap; here, Xj goes to a fixed point that depends on its initial value.
Fig 4.
The mean horizontal correlation coefficient 〈Rh〉 diagram for a set of situations mimicking various fly eyes according to Eq 9 in S1 Text.
The left half of the main diagram corresponds to β < 0 (Eq 5) while the right half corresponds to β > 0 (Eq 6). The horizontal axis shows the value of α which varies from 0 to 1 to the right and left of the y-axis. The vertical axis indicates the value of . The chromatic pink indicates 〈Rh〉 = −1, while the gray pixels show 〈Rh〉 = 0. Some examples of the simulated eye configurations are shown (panels (A)-(I)) for the following parameter values: (A) β > 0, α = 1. A value of β > 0 (Eq 6) indicates the Doli striped region, α = 1 makes the second term (
)on the right hand side of Eq 6 vanish, so that it becomes independent of
. We thus have a pure Doli configuration. (B) β > 0, α = 0.9 and
. This configuration deviates slightly from a perfect Doli eye by introducing some randomness via the Drosophila component. (C) β > 0, α = 0.65 and P0 = 0.9. As the value of α is reduced, the effect of the ordered Doli wanes, giving way to patterns that more closely resemble Drosophila. (D) β > 0, α = 0.65 and P0 = 0.1 Here we see Drosophila-like patterns, where the spatial distributions are random, although the ratio of the two colors is fixed. (E) α nearly zero,
and (F) α nearly zero,
: In both cases, a Drosophila dominant behavior is observed, with the different percentages of reds and greens corresponding to the different values of
. Note that for β < 0, i.e. the left half of the phase diagram, the effect of the so-called Uniform Fly increases as α increases at the expense of the Drosophila effect. The extreme case of this is at α = 1 when the configurations are a homogeneous green (or red). (G), (H) and (I): α = 0.9. with
(G),
(I) and
(H). The dominant behaviour is that of the Uniform Fly, which gives rise to a nearly uniform green(red) color. As
increases, red gives way increasingly to green.
Fig 5.
Inference of model parameters from simulated data.
(a) Average error in parameter inference in dependence of the true value α. (b) Average inferred value αinf as a function of true α. (c) Average inferred value ϵinf as a function of true ϵ. In panels (b) and (c), height of error bars indicates standard deviation; the dashed line is the diagonal, i.e. the identity mapping between true and average inferred values.