Figures and data in Investments in photoreceptors compete with investments in optics to determine eye design

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8 figures, 4 tables and 2 additional files

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Figure 1

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Eye structure and geometry define resolution and costs.

(a) Fused rhabdom apposition eye, photoreceptors coding a pixel form fused rhabdom and send axons to a single neural module; (b) neural superposition (NS) apposition eye, photoreceptor forms its own rhabdomere, photoreceptors with same optical axes code single pixel and send axons to single neural module; and (c) simple eye, as in a camera each photoreceptor codes a single pixel. (d) Gradient of investment in spatial acuity: apposition eye, honeybee drone, *Apis mellifera*. From ventral to dorsal, lens diameter $D$ increases and interommatidial angle $Δ ϕ$ decreases to increase spatial resolution, and rhabdom length, $L$ , increases to increase $S N R_{p h}$ . 10 μm thick longitudinal section, (DA) dorsal eye area and (VA) ventral area. BM – retina’s basement membrane; E – equator separating dorsal and ventral regions. From Menzel et al., 1991, original micrograph courtesy of Doekele Stavenga. (e) Schematic section of locally spherical apposition eye region. Volumes of optics and photoreceptor array are determined by dimensions that constrain the quality of the spatial image coded by photoreceptors: lens diameter $D$ , focal distance $f^{'}$ , interommatidial angle $Δ ϕ = D / R$ where $R$ is eye radius, and rhabdom(ere) length $L$ .

Figure 2

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The effect of trading investment in photoreceptor array for investment in dioptric apparatus.

The schematic shows two eye regions of equal specific volume, the left investing more heavily in photoreceptors, the right more heavily on optics. The images they capture show that transferring resources from photoreceptor array to optics increases image sharpness and contrast at the expense of increasing noise.

Figure 3

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The performance surface $H (D, L)$ describes how information capacity changes across all geometrically permissible configurations of an eye of fixed cost – the eye’s morphospace.

Red area: high-performance zone in which capacity is >95% maximum. Surfaces plotted for model fly neural superposition (NS) eyes, with F-number, $F = 2$ and total cost, $C_{tot} = 4 \times 10^{9}$ μm³ sr^–1 at three values of photoreceptor energy tariff $K_{E}$ . Acceptance angle $Δ ρ$ calculated using Snyder, 1979 CoG approximation. Data for all graphs is provided in Figure 3—source data 1. The Matlab code used to calculate performance surfaces across the morphospaces of the model eyes studied in this paper is publicly available at https://github.com/fjhheras/eyedesign,copy archived at Heras, 2025.

Figure 3—source data 1 Related to Figure 3a–e.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig3-data1-v1.xlsx
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Figure 4

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When model fly NS eyes are optimised for information capacity, eye structure, eye performance, and the division of resources between optics and photoreceptor array depend on total investment, $C_{t o t}$ and photoreceptor energy tariff, $K_{E}$ .

(a) Lens diameter $D$ ; (b) interommatidial angle, $Δ ϕ$ ; and (c) rhabdomere length, $L$ . Note the sudden jump in the curve for CoG $K_{E} = 0.12$ . (d) Optimum information capacity $H$ and rhabdomere length $L$ at CoG $K_{E} = 0.12$ plotted over the range of $C_{t o t}$ within which $L$ jumps upward. The eightfold increase in $L$ has very little effect on the rate of increase of $H$ . (e) $% C_{t o t}$ allocated to photoreceptor array. (f) Photoreceptor energy cost as $% C_{t o t}$ . CoG, acceptance angle approximated by convolving Gaussians (Snyder, 1979); WOM, acceptance angle approximated according to wave optics model (Stavenga, 2004a). Key in (b) also applies to (a). Data for all graphs is provided in Figure 4—source data 1.

Figure 4—source data 1 Related to Figure 4a–f.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig4-data1-v1.xlsx
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Figure 5

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Parameters defining the configurations of 12 fly NS eye regions, taken from 7 species (Table 2), compared to model NS eyes optimised for information capacity at given values of photoreceptor energy tariff $K_{E}$ , and total specific volume $V_{t o t}$ .

(a) Lens diameter $D$ , (b) interommatidial angle $Δ ϕ$ , (c) eye parameter $p = D Δ ϕ$ , and (d) rhabdomere length $L$ . (e) Log–log plot of $L$ vs $D$ , dashed line shows slope for isomorphic scaling, $L \propto D$ . (f) Specific volume of photoreceptor array, $V_{p h}$ expressed as % total specific volume of eye, $% V_{t o t}$ . Models use Snyder’s CoG approximation for photoreceptor acceptance angle (Snyder, 1979). Data for all graphs is provided in Figure 5—source data 1.

Figure 5—source data 1 Related to Figure 5a–f.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig5-data1-v1.xlsx
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Figure 6

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Theoretical results from apposition eye models optimised for information capacity compared to empirical data from 16 apposition eye regions taken from 3 species (Table 3).

(a) $D$ vs specific volume $V_{t o t}$ . (b) $Δ ϕ$ vs $V_{t o t}$ . (c) $L$ vs $V_{t o t}$ . (d) photoreceptor array specific volume $V_{p h}$ as $% V_{t o t}$ vs $V_{t o t}$ . (**e, f**) Models demonstrate impact of photoreceptor costs and its dependence on total cost, $C_{t o t}$ . (e) Photoreceptor cost, $C_{p h}$ , as $% C_{t o t}$ . (f) Photoreceptor energy cost as $% C_{p h}$ . Models run with F-number $F = 5.5$ , using COG approximation for acceptance angle (Snyder, 1979) and values of energy tariff $K_{E}$ given in keys. Data for all graphs is provided in Figure 6—source data 1.

Figure 6—source data 1 Related to Figure 6a–f.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig6-data1-v1.xlsx
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Figure 7

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Resource allocation in model simple eyes optimised for information capacity compared to optimised model neural superposition apposition eyes (NS).

(a) Schematic showing apposition eye and simple eye with identical spatial resolution, as defined by lens diameter $D$ , focal distance $f^{'}$ , rhabdom(ere) diameter $d_{r h}$ , and rhabdom(ere) length, $L$ (after Kirschfeld, 1976). Note the denser packing of photoreceptors inside the simple eye. (**b–e**) Properties of simple and NS eye models optimised for information capacity in full daylight. (b) Photoreceptor length, $L$ vs specific volume $V_{t o t}$ ; (c) photoreceptor specific volume $V_{p h}$ expressed as $% V_{t o t}$ vs $V_{t o t}$ . (d) Photoreceptor investment $C_{p h}$ as $% C_{t o t}$ vs $C_{t o t}$ . (e) photoreceptor energy cost as $% C_{t o t}$ vs $C_{t o t}$ . Models use Snyder, 1979 CoG approximation for acceptance angle. Data for all graphs is provided in Figure 7—source data 1.

Figure 7—source data 1 Related to Figure 7b–e.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig7-data1-v1.xlsx
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Figure 8

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Information rate, $H$ vs total cost $C_{t o t}$ for simple and apposition model eyes optimised to maximise information capacity in full daylight.

(a) Rates are higher in simple eyes and more sensitive to photoreceptor energy tariff, $K_{E}$ . (b) Efficiency, bits per unit specific volume per second, falls less steeply with increasing $C_{t o t}$ in simple eyes. Models use Snyder’s CoG approximation (1979) for acceptance angle. Data for all graphs is provided in Figure 8—source data 1.

Figure 8—source data 1 Related to Figure 8a, b.: https://cdn.elifesciences.org/articles/96517/elife-96517-fig8-data1-v1.xlsx
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Tables

Table 1

Symbols used in the Results and the Discussion.

Symbol	Description	Units
$D$	Lens diameter	μm
$f$	Lens focal length, in air	μm
$f^{'}$	Focal distance	μm
$F$	F-number
$d_{r h}$	Diameter of rhabdom (fused-rhabdom eye) or rhabdomere (NS eye)	μm
$Δ ϕ$	Interommatidial or interreceptor angle	rad (equations)
		° (figures)
$R$	Radius of eye or locally spherical eye region	μm
$p$	Eye parameter
$L$	Length of rhabdom or rhabdomere, also depth of photoreceptor array	μm
$λ$	Wavelength of light	nm
$η_{i}$	Refractive index of eye’s internal medium
$Δ ρ_{l}$	Half width of lens point-spread function	rad
$Δ ρ_{r h}$	Half-width of rhabdom(ere) acceptance angle	rad
$Δ ρ_{p h}$	Half-width of photoreceptor acceptance angle	rad
$V_{o}$	Volume of optics	μm³ sr⁻¹
$V_{p h}$	Volume of photoreceptor array	μm³ sr⁻¹
$V_{t o t}$	Total eye volume	μm³ sr⁻¹
$C_{o}$	Cost of optics	μm³ sr⁻¹
$C_{p h}$	Costs of photoreceptor array	μm³ sr⁻¹
$C_{t o t}$	Total cost of eye	μm³ sr⁻¹
$S_{E}$	Photoreceptor energy surcharge	μm³ sr⁻¹
$K_{E}$	Photoreceptor energy tariff	μm³ per microvillus
$N_{v i l}$	Number of microvilli
$ν$	Number of microvilli per unit length of rhabdom or rhabdomere	μm⁻¹
$ψ$	Transduction rate	s⁻¹
$S N R_{p h}$	Photoreceptor signal to noise ratio	per unit contrast
$H$	Spatio-temporal information capacity	bits sr⁻¹ s⁻¹

Table 2

Measurements of Dipteran neural superposition (NS) eyes, extracted from published sources (Land, 1997; Hardie, 1985; Stavenga et al., 1990; Stavenga, 2003a; Land and Eckert, 1985; Stavenga, 2003b; Gonzalez-Bellido et al., 2011; Zeil, 1983; Wardill et al., 2017).

Further details of measurements made, their use and results obtained are given in Supplementary file 1.

Species	Eye region	D (µm)	$Δ ϕ$ (°)	L (µm)	f (µm)	f’ (µm)	F	Source
Calliphora vicina	Male acute zone	37	1.07	340	74	99	2	Land, 1997; Hardie, 1985; Stavenga et al., 1990; Stavenga, 2003a
Calliphora vicina	Female acute zone	29	1.28	280	58	78	2	Hardie, 1985
Musca domestica	Male love spot	36	1.75	250	72	96	2	Hardie, 1985; Land and Eckert, 1985
Musca domestica	Male peripheral	20	3.5	140	40	54	2	Hardie, 1985
Drosophila melanogaster	Average	16.5	5	83	20	27	1.25	Land, 1997; Hardie, 1985; Stavenga, 2003b
Coenesia attenuata	Female frontal	20	2.3	140	25	28	1.25	Gonzalez-Bellido et al., 2011
Bibio marci	Male dorsal	33	1.6	230	70	94	2.1	Zeil, 1983
Bibio marci	Male ventral and female	21	3.7	110	36	48	1.7	Zeil, 1983
Dilophus febrilis	Male dorsal	24	2.2	180	40	54	1.7	Zeil, 1983
Dilophus febrilis	Male ventral and female	15	5.1	60	16	21	1.1	Zeil, 1983
Holcocephela fusca	Acute zone, max. acuity	75	0.28	230	160	176	2.53	Wardill et al., 2017
Holcocephela fusca	Peripheral	21	3.5	123	33	45	1.6	Wardill et al., 2017

Table 3

Measurements of fused-rhabdom apposition eyes, extracted from published sources (Rossel, 1979; Labhart and Nilsson, 1995; Menzel et al., 1991; Varela and Wiitanen, 1970; Kelber and Somanathan, 2019).

Further details of measurements made, their use and results obtained are given in Supplementary file 1.

Species	Eye region	D (µm)	$Δ ϕ$ (°)	L (μm)	f (μm)	f’ (μm)	F	Source
Tenodora australasiae	Eye coordinates z = 15; x = 20 (fovea)	47.5	0.64	500	205	445	8	Rossel, 1979
	z = 15; x = 30	51.5	0.8	535	335	405	6.5	Rossel, 1979
	z = 15; x = 10	41	1	415	305	375	7.4	Rossel, 1979
	z = 15; x = 0	35	1.1	285	205	275	5.9	Rossel, 1979
	z = 15; x = 60	42	1.6	390	150	212	3.6	Rossel, 1979
	z = 15; x = 100	33	2.5	330	110	153	3.3	Rossel, 1979
Sympetrum spp.	Ventral –60°	31	1.85	461	124	166	4	Labhart and Nilsson, 1995
	Ventral –5°	27	1.75	461	108	148		Labhart and Nilsson, 1995
	Dorso-ventral border	31	1.69	516	124	167		Labhart and Nilsson, 1995
	Dorsal +50	53	1.82	627	92	123		Labhart and Nilsson, 1995
	Dorsal +70	61.5	1.78	886	246	330	4	Labhart and Nilsson, 1995
	Dorsal fovea	71	0.35	1107	305	410	4.4	Labhart and Nilsson, 1995
Apis mellifera drone	Dorsal eye	40	1	500	205	275	5.1	Menzel et al., 1991
	Top ventral	28	2.1	400	145	194	5.1	Menzel et al., 1991
	Mid ventral	25	2.8	280	82	110	3.3	Menzel et al., 1991
	Basal ventral	18	4.4	150	54	72	3	Menzel et al., 1991
Apis mellifera worker	Max. acuity	25	1.6	350	100	134	5.6	Varela and Wiitanen, 1970; Kelber and Somanathan, 2019

Table 4

Dependence of energy surcharge $K_{E}$ on time spent flying $T_{F}$ and hours of daylight $D_{L}$ , calculated for blowfly.

T_F (hr)	D_L (hr)	K_E (µm³/microvillus)
12	12	0.13
12	16	0.15
2	12	0.44
2	16	0.52

Additional files

Supplementary file 1 Annotated spreadsheet of the anatomical and optical data extracted from the literature that was used to provide the values given in Tables 2 and 3, and plotted in Figures 5 and 6.: https://cdn.elifesciences.org/articles/96517/elife-96517-supp1-v1.xlsx
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Francisco JH Heras
Simon B Laughlin

(2026)

Investments in photoreceptors compete with investments in optics to determine eye design

eLife 13:RP96517.

https://doi.org/10.7554/eLife.96517.3

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Eye structure and geometry define resolution and costs.

The effect of trading investment in photoreceptor array for investment in dioptric apparatus.

The performance surface H(D,L) describes how information capacity changes across all geometrically permissible configurations of an eye of fixed cost – the eye’s morphospace.

Figure 3—source data 1

When model fly NS eyes are optimised for information capacity, eye structure, eye performance, and the division of resources between optics and photoreceptor array depend on total investment, Ctot and photoreceptor energy tariff, KE.

Figure 4—source data 1

Parameters defining the configurations of 12 fly NS eye regions, taken from 7 species (Table 2), compared to model NS eyes optimised for information capacity at given values of photoreceptor energy tariff KE, and total specific volume Vtot.

Figure 5—source data 1

Theoretical results from apposition eye models optimised for information capacity compared to empirical data from 16 apposition eye regions taken from 3 species (Table 3).

Figure 6—source data 1

Resource allocation in model simple eyes optimised for information capacity compared to optimised model neural superposition apposition eyes (NS).

Figure 7—source data 1

Information rate, H vs total cost Ctot for simple and apposition model eyes optimised to maximise information capacity in full daylight.

Figure 8—source data 1

Symbols used in the Results and the Discussion.

Measurements of Dipteran neural superposition (NS) eyes, extracted from published sources (Land, 1997; Hardie, 1985; Stavenga et al., 1990; Stavenga, 2003a; Land and Eckert, 1985; Stavenga, 2003b; Gonzalez-Bellido et al., 2011; Zeil, 1983; Wardill et al., 2017).

Measurements of fused-rhabdom apposition eyes, extracted from published sources (Rossel, 1979; Labhart and Nilsson, 1995; Menzel et al., 1991; Varela and Wiitanen, 1970; Kelber and Somanathan, 2019).

Dependence of energy surcharge KE on time spent flying TF and hours of daylight DL, calculated for blowfly.

Supplementary file 1

MDAR checklist

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The performance surface $H (D, L)$ describes how information capacity changes across all geometrically permissible configurations of an eye of fixed cost – the eye’s morphospace.

When model fly NS eyes are optimised for information capacity, eye structure, eye performance, and the division of resources between optics and photoreceptor array depend on total investment, $C_{t o t}$ and photoreceptor energy tariff, $K_{E}$ .

Parameters defining the configurations of 12 fly NS eye regions, taken from 7 species (Table 2), compared to model NS eyes optimised for information capacity at given values of photoreceptor energy tariff $K_{E}$ , and total specific volume $V_{t o t}$ .

Information rate, $H$ vs total cost $C_{t o t}$ for simple and apposition model eyes optimised to maximise information capacity in full daylight.

Dependence of energy surcharge $K_{E}$ on time spent flying $T_{F}$ and hours of daylight $D_{L}$ , calculated for blowfly.