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

In daylight, when the number of available photons is not limiting, investments in optics buy increased optical contrast and spatial resolving power by reducing optical and geometrical constraints, as follows. An apposition eye forms and captures images with an array of ommatidia (Figure 1). Each ommatidium uses a facet lens of diameter, $D$, and focal length (in air), $f$, to focus light onto the entrance aperture of photoreceptive waveguides, a single fused rhabdom or several rhabdomeres (Figure 1) of diameter $d_{rh}$ . Increasing $D$ sharpens a photoreceptor’s angular sensitivity by reducing the blur produced by the Airy disk diffraction pattern, whose half-width $\mathrm{\Delta }{\rho }_l=\lambda /D$ radians, where $\lambda$ is the wavelength of light, taken to be 500 nm. Increasing $f$ and reducing $d_{rh}$ also sharpen angular sensitivity by reducing the angular subtense of the photoreceptor entrance aperture, $d_{rh}/f$. Sharpening angular sensitivity has two benefits: it increases spatial resolving power and increases image contrast.

We calculate the effects of $D$, $d_{rh}$, and $f$ on the half-width of a fly photoreceptor’s angular sensitivity, $\mathrm{\Delta }\rho$, using two formulae, Snyder’s simplification, his convolution of Gaussians (CoG) $\mathrm{\Delta }{\rho }^2=\mathrm{\Delta }{\rho }_l^2+\mathrm{\Delta }{\rho }_{rh}^2$, where $\mathrm{\Delta }{\rho }_l=\lambda /D$ and $\mathrm{\Delta }{\rho }_{rh}=d_{rh}/f$; and Stavenga’s better approximation, $\mathrm{\Delta }\rho =1.26\mathrm{\Delta }{\rho }_l$, obtained using a comprehensive wave-optics model (WOM) of flies’ NS eyes operating in bright light (see Methods for more details). For CoG, we fix $d_{rh}$ = 1.9 μm because this value is typical of fly R1–6 (Hardie, 1985). CoG provides a lower bound to optical resolving power and WOM a more efficient upper bound that, in bright light, makes better use of investments in optics by operating closer to the diffraction limit.

The benefits of investing in optics come at a cost: increasing $D$ and lengthening $f$ to reduce $\mathrm{\Delta }\rho$ expands the volume of the dioptric apparatus, $V_o$, which in apposition eyes is easily calculated from eye geometry (Figure 1e). $D$ and $\mathrm{\Delta }\varphi$ define the radius $R=D/\mathrm{\Delta }\varphi$ of a locally spherical eye region, in which the dioptric apparatus (lens, cone, and screening pigment) is a shell of thickness $f^{\mathrm{\prime }}$, the focal distance from lens front surface to focal plane. Thus,

Because $V_o$ is a specific measure, volume per unit solid angle of visual space, we can compare investments in eye regions that differ in radius and angular extent. Thus, $V_o$ handles the regional variations in $R$ observed in many compound eyes.

To simplify our expression for $V_o$ (Equation 1) we fix lens F-number, $F=f/D$ and apply the formula for image formation by a convex cornea, $f^{\mathrm{\prime }}=f.n_i$, where $n_i$ is the refractive index of the internal medium (Nilsson and Land, 2012). This formula describes several apposition eyes, including fly NS eyes (Rossel, 1979; Stavenga et al., 1990; Stavenga, 2003a; Zeil, 1983). Thus,

We use values measured in blowfly, $F=2.0$ (Stavenga, 2003a) and $n_i$, = 1.34 (Seitz, 1968).

Fixing $F$ both simplifies our models by reducing the number of free parameters and accounts for the dependence of the photon flux entering an ommatidium on $D$ by making image brightness (the flux per unit area of rhabdomere entrance aperture) independent of $D$. We use the specific volume of optics, $V_o$, as our measure of the cost of optics, $C_o$ μm³ sr⁻¹. This usage assumes that the cost of materials, the cost of metabolic energy for maintaining function, and the energy cost of carriage all increase in equal proportion to volume, in all parts of the dioptric apparatus, namely the corneal lens, cone, and surrounding pigment cells. Given the lack of definitive data on composition and costs, this assumption is a reasonable starting point (Discussion).

Investing in photoreceptors buys photoreceptor signal-to-noise ratio, $SN{R}_{ph}$, by reducing the effects of an optical constraint, the Poisson statistics of photon absorption (Nilsson and Land, 2012), and at higher light levels a physiological constraint, the saturation of a photoreceptor’s transduction units – its light sensitive microvilli (Heras and Laughlin, 2017; Howard et al., 1987; Song et al., 2012). A microvillus contains rhodopsin molecules, the intermediate molecules of the phototransduction cascade and the ion channels they activate, and it uses these to generate a brief pulse of depolarising current, a quantum bump, following the activation of a single rhodopsin molecule by an absorbed photon. This all-or-none response takes time to complete, during which the microvillus cannot respond to the absorption of another photon (Hardie and Postma, 2008; Song and Juusola, 2014). Therefore, as light intensity rises to daylight levels, $SN{R}_{ph}$ falls below the Poisson limit because a photoreceptor’s transduction units saturate (Howard et al., 1987). At any given time, a significant fraction of microvilli are failing to respond to the absorption of a photon by rhodopsin because they are already engaged in processing a bump. Consequently, the transduction rate $\psi$ no longer increases linearly with the absorption rate, and bump statistics change from Poisson to binomial. In daylight, flies activate a photomechanical response, a longitudinal pupil, to prevent excessive saturation and maintain $SN{R}_{ph}$ close to the maximum permitted by binomial statistics,

where $SN{R}_{ph}$ is defined with respect to a signal of unit contrast (Howard et al., 1987). Note that by operating at this upper limit, the fly obtains maximum benefit from investing in $N_{vil}$. Thus, the benefit of investing in a longer rhabdomere is an increase in $SN{R}_{ph}$ in bright light, according to

where $L$ is rhabdomere length and $\nu$ is the number of microvilli per unit length. We assume $\nu$ is constant along the rhabdomere and from published results we estimate $\nu =230{\mathrm{\mu }\mathrm{m}}^{-1}$ 230 µm⁻¹ (Methods). We acknowledge that diameter, cross section, and taper often vary within and among rhabdomeres. We are fixing $\nu$ at a plausible value for a generic model.

The cost of increasing $SN{R}_{ph}$ by increasing $L$ is easily calculated. In fly NS eyes, $L$ is also the depth of the photoreceptor array because the rhabdomeres of the six largest photoreceptors, R1–6, stretch from the focal plane of the lens to the basement membrane (Hardie, 1985). Thus in a locally spherical eye region (Figure 1) the photoreceptor array’s specific volume is

Substituting $f^{\mathrm{\prime }}=F.n_i.D$

According to eye geometry (Figure 1), the total specific volume of an eye region is

Observe that because both $V_o$ (Equations 1 and 2) and $V_{ph}$ (Equation 5; Equation 6) depend upon $R$ and $R=D/\mathrm{\Delta }\varphi$, both $V_o$ and $V_{ph}$ increase with the number of ommatidia per unit solid angle.

For the photoreceptor array, we assume that the costs of space, materials and energy for carriage increase in proportion to volume with the same constants of proportionality as optics. Again, in the absence of measurements, this assumption is a reasonable starting point (Discussion). However, this assumption breaks down when we consider specific metabolic rates. To enable and power phototransduction, a photoreceptor has an exceptionally high specific metabolic rate (energy consumed per gram, and hence unit volume, per second) (Laughlin et al., 1998; Niven et al., 2007; Pangršič et al., 2005). We account for this extra cost by applying an energy surcharge, $S_E$. To equate with our other estimates of cost, this surcharge has units of μm³ sr⁻¹, and we assume that it increases in proportion to $N_{vil}$ because $N_{vil}$ determines the magnitude of the photoreceptor’s light-gated conductance (Hardie and Postma, 2008; Heras and Laughlin, 2017). We satisfy these two requirements by defining

where $K_E$, the photoreceptor energy tariff, converts the energy consumed by a photoreceptor per microvillus into an equivalent volume.

We estimate $K_E$ (Methods) by adopting the method used to estimate the energy cost of weapons carried by flying beetles (Goyens et al., 2015). We divide the energy consumed per microvillus by the animal’s mass specific metabolic rate to obtain an equivalent body mass which, assuming a density of 1.0, is our volume equivalent, $K_E{\mathrm{\mu }\mathrm{m}}^3$ per microvillus. $K_E$ is at present poorly determined: it depends upon several ecological, physiological and behavioural factors, it will vary among species and among individuals of the same species, and there is barely enough data (Methods). Given this uncertainty, we model a range of values from $K_E$ = 0 to $K_E$ = 0.64 that covers the range we estimate for blowfly, $K_E$ = 0.13 to $K_E$ = 0.52 (Methods).

To summarise costs, for the photoreceptor array

and for the optics

Because $C_o$ and $C_{ph}$ have the same units and are specific (per unit solid angle), we can transfer resources between optics and photoreceptors within the constraint of total specific cost

When $C_{tot}$ is fixed, allocating a smaller proportion to optics and a larger proportion to photoreceptors, trades the benefits of investing in optics for the benefits of investing in photoreceptors (Figure 2).

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Having used constraints to link investments in dioptrics and photoreceptor array to three major determinants of the quality of the achromatic images captured by fly photoreceptors R1–6, namely $\mathrm{\Delta }\rho$, $\mathrm{\Delta }\varphi$, and $SN{R}_{ph}$, we can now model how trade-offs between optics and photoreceptors change an eye’s ability to support vision. Several measures of visual performance depend on $\mathrm{\Delta }\rho$, $\mathrm{\Delta }\varphi$, and $SN{R}_{ph}$. To establish proof of principle, we use a general measure that embraces a wide variety of resolvable achromatic image details – spatio-temporal information capacity in bits sr⁻¹ s⁻¹. Information capacity has proved useful for measuring the performance of compound eyes (de Ruyter van Steveninck and Laughlin, 1996), discovering design principles (van Hateren, 1992; Howard and Snyder, 1983; Snyder et al., 1977), and illuminating the evolution of simple eyes (Nilsson and Pelger, 1994). The measure is particularly relevant for a fly NS eye because coding by second order neurons is adapted to maximise information capacity (van Hateren, 1992; Laughlin, 1981).

To model the effects of resource allocation on information capacity, we fix $C_{tot}$ and generate pairs of $D$ and $L$ that cover the range of values permitted by eye geometry. Each $D,L$ pair specifies an investment in optics, $C_o(D,L)$, an investment in photoreceptors $C_{ph}(D,L)$ and hence the three determinants of image quality, $\mathrm{\Delta }\rho (D,L)$, $\mathrm{\Delta }\varphi (D,L)$, and $SN{R}_{ph}(D,L)$, that specify spatio-temporal information capacity $H(D,L)$ bits sr⁻¹ s⁻¹.

We calculate $H(D,L)$ (Methods) in the frequency domain (Howard and Snyder, 1983; van Hateren, 1992). In brief, a 2D power spectrum typical of natural scenes is low-pass filtered by the photoreceptor angular sensitivity function, $\mathrm{\Delta }\rho (D,L)$, and sampled by a hexagonal lattice of photoreceptors specified by $\mathrm{\Delta }\varphi (D,L)$. Movement of the retinal image modulates the flux of photons entering a photoreceptor by converting spatial frequencies into temporal frequencies, according to the distribution of angular velocities generated during behaviour. Then, during transduction, the photoreceptors low-pass filter these temporal frequencies and add the noise generated by sampling the photon flux stochastically with transduction units (microvilli). We use (van Hateren, 1992) formulae for the $1/f^2$ power spectrum of natural scenes and the distribution of image velocities appropriate for blowfly, and we temporally low-pass filter according to the measured properties of a fully light-adapted blowfly photoreceptor R1–6 (Methods). We also modify van Hateren’s method to take account of the spatial aliasing that occurs when an array of ommatidia undersamples an image (Methods).

By calculating information capacities for each $D,L$ pair, we define the performance surface $\mathit{H}\mathbf{(}\mathit{D}\mathbf{,}\mathit{L}\mathbf{)}$ that covers the morphospace of model eyes of given F-number, $F$, and specific cost, $C_{tot}$. The performance surface is a flat-topped ridge with steep flanks (Figure 3). Atop the ridge, a single point of maximum capacity, $H(D_{opt},L_{opt})$, sits in an extensive zone within which capacity, and hence efficiency, is $>95\mathrm{\%}$ maximum (red zone in Figure 3). With efficiency dropping steeply away from the ridge, approximately two thirds of $D,L$ combinations have efficiencies $<70\mathrm{\%}$.

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The shape of the performance surface depends upon photoreceptor energy cost (Figure 3). Increasing the energy tariff $K_E$ reduces $L_{opt}$ from the maximum permitted by eye geometry to one quarter of maximum. However, because the ridge is flat-topped, $L_{opt}$ still lies in an extensive high-efficiency zone, within which $L$ can be changed threefold and $D$ by 40%, while maintaining $>95\mathrm{\%}$ efficiency (Figure 3). This broad high-efficiency zone is bad news for investigators who set out to establish that eyes are optimised, but good news for insects. An eye region can be adapted for a specific purpose, for example by increasing $L$ to increase SNR, without sacrificing more than 5% of a measure of general-purpose vision, information capacity. Thus, this robust region of design space can facilitate evolution by reducing loss of function.

We next establish how, in theory, eye structure and investments in optics and photoreceptor array vary with total investment when efficiency is optimised by maximising information capacity. We run our fly NS eye model with different combinations of $C_{tot}$ and $K_E$ and for each combination find the values of $L_{opt}$, $D_{opt}$, and $\mathrm{\Delta }{\varphi }_{opt}$ that maximise information capacity and the corresponding investments in optics and photoreceptor array, $C_o$ and $C_{ph}$. For each combination of $C_{tot}$ and $K_E$ we run two versions of our NS model, one calculates $\mathrm{\Delta }\rho$ using CoG (Snyder, 1979) and the other WOM, which produces a narrower $\mathrm{\Delta }\rho$ that comes closer to the lens diffraction limit (Stavenga, 2004a). The two versions give similar patterns of investment (Figure 4).

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Our new method for calculating costs enables us to estimate, for the first time, the total investment an animal makes in an eye or eye region, and break this down into its two major components, optics and photoreceptor array. Using published measurements of $D$, $\mathrm{\Delta }\varphi$, and $L$ we determine the specific volumes of dioptric apparatus, $V_o$, and photoreceptor array, $V_{ph}$ in fly NS eyes. Although numerous publications give values of $D$ and $\mathrm{\Delta }\varphi$ (Land, 1997; Feller et al., 2021), very few include $L$. We have values for just 12 eye regions, taken from 7 species of flies (Table 2). For each region, we insert the values, $L$, $D$, $\mathrm{\Delta }\varphi$, and $f^{\mathrm{\prime }}$, into Equations 1 and 5 to calculate $V_o$, $V_{ph}$, and $V_{tot}$. Where focal distance $f^{\mathrm{\prime }}$ is not reported (Table 2), we follow our models (Equation 2) and calculate $f^{\mathrm{\prime }}=FD{n}_i$ with $F=2$ and $n_i=1.34$.

The 12 eye regions provide a well-distributed set of empirical values of $L$, $D$, $\mathrm{\Delta }\varphi$, $V_o$, $V_{ph}$, and $V_{tot}$ that spans almost five orders of magnitude of $V_{tot}$, from $1.4\times {10}^6$ μm³ sr⁻¹ and $2.1\times {10}^6$ μm³ sr⁻¹for the low acuity eye region of the March fly Dilophus febrilis and the eye of $9\times {10}^{10}$ μm³ sr⁻¹ Drosophila melanogaster, respectively, to for the remarkable fovea of the small robber fly Holcocephala fusca (Wardill et al., 2017).

We compare the empirical values obtained from flies with theoretical curves, obtained from model NS eyes that are optimised for information capacity (Figure 5). Note that for our theoretical curves, we plot $D$, $\mathrm{\Delta }\varphi$, $L$, and $V_{ph}$ against total specific cost, $C_{tot}{\mathrm{\mu }\mathrm{m}}^3{\mathrm{sr}}^{-1}$ (Figure 4). This total includes the surcharge for photoreceptor energy consumption $S_E{\mathrm{\mu }\mathrm{m}}^3{\mathrm{sr}}^{-1}$, which increases in proportion to the number of microvilli and the photoreceptor energy tariff, $K_E$ (Equation 8). However, we cannot estimate $C_{tot}$ in flies because their values of $K_E$ are not well-determined (Methods). Although $K_E$ will vary considerably between species, we only have enough published data to calculate its order of magnitude and likely range (Methods). Furthermore, $K_E$ is species-specific and, because we have no more than two data points per species and we are using a generic model that does not account for differences in $d_{rh}$ and $F$ among species (Table 2), we cannot estimate $K_E$ by fitting our model to the empirical data. For these reasons, we compare theoretical findings to empirical data by plotting them against a measure of eye size that is both calculated by our models and estimated from data, total specific volume, $V_{tot}$. Because $K_E$ is uncertain, we plot theoretical curves calculated with values of $K_E$ from within our likely range.

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Looking across 12 fly eye regions, $D$ increases and $\mathrm{\Delta }\varphi$ decreases as $V_{tot}$ increases. $D$ lies within, or close to, the theoretical envelope defined by CoG models optimised with $K_E$ = 0 and $K_E$ = 0.32 (Figure 5a), and tends to slightly exceed optimum values at lower specific volumes. $\mathrm{\Delta }\varphi$ is approximately 25% wider than optimum (Figure 5b) in the four eye regions with lowest specific volume and close to optimum elsewhere. The eye parameter, $p=D\mathrm{\Delta }\varphi$ (Figure 5c) indicates the degree to which an apposition compound eye sacrifices spatial resolution by sampling below the lens diffraction limit. Only Holcocephala fovea, $p=0.37$, comes close to the diffraction limit $p=0.29$ (Snyder, 1979). As observed in a number of diurnal apposition eyes (Wehner, 1981), the remaining 11 fly eye regions in our data set undersample by a factor of 2 or more, with $p$ decreasing with increasing $V_{tot}$ from 1.4 to 0.65 (Figure 5c). Our optimised models also undersample, with $p$ decreasing with increasing $V_{tot}$ from 0.89 to 0.65 (Figure 5c). Note that like $D$ and $\mathrm{\Delta }\varphi$, $p$ is relatively insensitive to $K_E$. We conclude that in most eye regions the eye parameter $p$ is not optimised for spatio-temporal information capacity, at least according to our model, because it is higher than predicted at all but the highest values of $V_{tot}$.

Rhabdomere length, $L$, increases with $V_{tot}$ from 60 μm to 340 μm (Figure 5d; Table 2), within or close to the theoretical envelope defined by $K_E=0$ and $K_E=0.16$. There is a robust trend: the empirical values of log $L$ increase with log $V_{tot}$, and hence with increasing $D$ and decreasing $\mathrm{\Delta }\varphi$ (Figure 5a, b), along a trajectory that resembles theoretical curves for optimum log $L$. However, the data do not follow the theoretical curve for any particular value of $K_E$, and there is some scatter among the data points (Figure 5d). We have three reasons to attribute most of this scatter to unknown differences in $K_E$. First, the optimum value of $L$ is sensitive to $K_E$ (Figure 4c); second, $K_E$ will vary between species (Methods); and third, the empirical values of $D$ and $\mathrm{\Delta }\varphi$, which are in theory relatively insensitive to $K_E$, are less scattered and lie closer to our theoretical predictions (Figure 5a, b). The one obvious outlier, the shorter than predicted rhabdomeres in Holcocephala fovea (Figure 5d), is easily explained: the optics and photoreceptor array of this forward pointing fovea, together with underlying optic lobe, are tightly packed into a head that is less than 600 μm from front to back (Wardill et al., 2017), and this lack of headroom limits the depth of the eye, ($L+f^{\mathrm{\prime }}$).

The log–log plot of $L$ vs $D$ (Figure 5e) shows that the increase in $L$ with $V_{tot}$ is not explained by proportional scaling because the empirical values do not lie on a straight line of slope 1, $L=\mathrm{k}D$. As expected of a physiological system subject to several competing constraints (Taylor and Thomas, 2014), the theoretical curve continuously changes slope, and this is reflected in the data. Thus, the matching of $L$ to $D$ and $\mathrm{\Delta }\varphi$ observed in the data is not the consequence of a simple developmental constraint; it is the product of developmental mechanisms that allocate resources efficiently to optics and photoreceptor array.

We conclude that the eye regions we analyse are, according to our generic model, not optimised for spatio-temporal information capacity because the data points that are least sensitive to the photoreceptor energy tariff $K_E$, namely $D$, $\mathrm{\Delta }\varphi$ and $p$, are generally larger than predicted. Nonetheless, we argue on two grounds that the matching of rhabdomere length to spatial resolution seen in flies promotes the efficiency of their total investments in eye. First, $L$ lies within, or close to, the theoretical envelope predicted by models optimised for efficiency and second, because the theoretical performance surface has a wide flat high-efficiency zone (Figure 3), large deviations from optimum have relatively little impact on efficiency.

To accommodate photoreceptors’ lengthy rhabdomeres, flies allocate between 56% and 78% of eye volume, $V_{tot}$, to photoreceptor arrays (Figure 5f). The smaller remainder, between 44% and 22%, is allocated to optics. There is a slight trend for the percentage allocated to photoreceptors to increase with $V_{tot}$. With the exception of Holcocephala’s fovea, where the depth of the photoreceptor array is much less than predicted for optimum allocation, the volumes allocated to photoreceptor arrays lie within or close to the theoretical envelope defined by $K_E=0$ and $K_E=0.16$. On this basis, we conclude that our diurnal flies allocate a larger fraction of eye volume to photoreceptors than to optics, and this allocation promotes the efficient use of resources. Thus, photoreceptor costs are playing a major role in determining the design and efficiency of flies’ NS eyes.

We investigate diurnal apposition eyes with fused rhabdoms by modifying our NS eye model and comparing theoretical results with empirical data using the procedures developed for NS eyes. The modifications are straightforward. We increase lens F-number to 5.5 and reduce $d_{rh}$ to 1.8μm. These values are in the middle of the wide range found in our sample of apposition eye regions (Table 3). We also adjust the relationship between $L$ and the number of microvilli $N_{vil}$ to account for a fused rhabdom (Methods). For simplicity, we assume that all photoreceptors contribute equal numbers of microvilli along the full length of the rhabdom so that $N_{vil}$ increases in proportion to the depth of the photoreceptor array.

We then compare the models’ predictions of the optimum values of $L,D,\mathrm{\Delta }\varphi ,V_o,V_{ph}$, and $V_{tot}$ with empirical values extracted from published measurements of $D,\mathrm{\Delta }\varphi ,L$, and $f^{\mathrm{\prime }}$ (Table 3). The necessary data is only available from studies of four species: the mantid Tenodera australis (Rossel, 1979), the honeybee Apis mellifera (Kelber and Somanathan, 2019; Menzel et al., 1991; Varela and Wiitanen, 1970), and two species of the Libellulid dragonfly Sympetrum that are so similar that their data are lumped together (Labhart and Nilsson, 1995). The studies provide empirical values for 16 eye regions, which span a range of $V_{tot}$, from $4.4\times {10}^7$ μm³ sr⁻¹ for Apis worker ventral eye to $1.8\times {10}^{11}$ μm³ sr⁻¹ for Sympetrum dorsal fovea (Figure 6).

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Looking across our limited sample of fused rhabdom apposition eyes, $D$ increases with $V_{tot}$, $\mathrm{\Delta }\varphi$ decreases, and $L$ increases from 280 μm to 1100 μm (Figure 6a–c). These empirical values are more widely scattered than those for NS eyes (Figure 5a, b, d), and the discrepancies between empirical and predicted values are larger. We expect a wider scatter because our sample of fused rhabdom eyes is more diverse than our sample of NS eyes, both structurally and phylogenetically. In addition, the scatter is dominated by regional differences within eyes (Figure 6), several of which adapt a region for a specific task. For example, the 1100 μm rhabdomeres of the dragonfly Sympetrum, which are the longest photoreceptive waveguides found in any eye, adapt the fovea for detecting small prey against a bright blue sky by enhancing $SN{R}_{ph}$ (Labhart and Nilsson, 1995). Despite the scatter in data points, fused rhabdom eyes follow the trends seen in NS eyes (Figure 6 cf. Figure 5). In most eye regions, $D$ and $\mathrm{\Delta }\varphi$ are larger than predicted by models optimised for efficiency (Figure 6a, b), so therefore is the eye parameter $p$ (values not plotted). At low $V_{tot}$, $L$ lies within the theoretical envelope defined by models run with plausible values of photoreceptor energy tariff, $K_E$. However, at higher specific volumes $L$ falls increasingly short (Figure 6c). As with flies’ NS eyes, large investments are made in photoreceptor arrays. The volume fractions of photoreceptor arrays increase from 33% $V_{tot}$ to 75% $V_{tot}$ as $V_{tot}$ increases, and the majority of values lie between 40% $V_{tot}$ and 60% $V_{tot}$ (Figure 6d). Our information maximisation models predict an increase from 33% $V_{tot}$ to 85% $V_{tot}$.

Our models also give the relative contribution of photoreceptor cost to total cost when fused rhabdom apposition eyes are optimised for spatial information capacity. The percentage of total investment devoted to the photoreceptor array increases with the energy tariff $K_E$ and climbs steadily with increasing $C_{tot}$, from 43% to more than 80% (Figure 6e). As in NS eyes, the contribution of photoreceptor energy consumption to array cost is progressively marginalised by array volume cost as $C_{tot}$ increases, falling to less than 20% in the largest eyes (Figure 6f).

We conclude that the 16 eye regions in our sample of fused rhabdom apposition eyes are not optimised for spatio-temporal information capacity, at least according to our models. However, the regions exhibit three trends that are associated with efficient resource allocation – rhabdomeres are relatively long, length increases with acuity, and the photoreceptor array occupies a large fraction of eye volume. These trends suggest that, as in NS eyes, resources are being divided between optics and photoreceptor array to promote efficiency. Consequently, photoreceptor costs play a major role in the design of fused-rhabdom apposition eyes.

To further investigate how investment patterns differ according to eye type, we construct a basic hemispherical model of a simple eye with rhabdomeric photoreceptors (Methods; Figure 7a) that captures the equivalence of $D$, $f^{\mathrm{\prime }}$, and $\mathrm{\Delta }\varphi$ in apposition and simple eyes (Kirschfeld, 1976). We then follow the procedures described above to see how optimum patterns of investment depend on total investment, $C_{tot}$, and photoreceptor energy tariff, $K_E$. We consider the same 5 orders of magnitude of total investment, $C_{tot}=1\times {10}^6$ to $2.5\times {10}^{11}$ μm³ sr⁻¹, and for simple eye models this encompasses a 100-fold range of lens diameters, $D=$ 29 μm to 2.9 mm, and focal distances, f′ = 79 µm to 7.9 mm.

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For each $C_{tot}$, there is an optimum division of resources between optics and photoreceptors that maximises information capacity. $L_{opt}$ increases with $V_{tot}$, scaling close to ${\left(C_{tot}\right)}^{1/3}$ (Figure 7b). Increasing the energy tariff, $K_E$, greatly reduces $L_{opt}$ without changing this scaling exponent. At any given $C_{tot}$ and $K_E$, $L_{opt}$ is much shorter in simple eyes and the difference increases with $C_{tot}$ from at least 50% shorter at $C_{tot}=1\times {10}^6$ μm³ sr⁻¹ to over 90% shorter at $C_{tot}=1\times {10}^9$ μm³ sr⁻¹. We conclude that, in theory, efficient resource allocation explains why photoreceptors in diurnal simple eyes have much shorter light-sensitive waveguides than photoreceptors in diurnal apposition eyes.

Patterns of investment differ from apposition eyes in other respects (Figure 7). In simple models, the optimum volume fraction of the photoreceptor array is insensitive to $C_{tot}$ whereas in apposition models, it increases with $C_{tot}$. In addition, a simple eye is much more sensitive to photoreceptor energy consumption. The small energy tariff, $K_E=0.04$, reduces simple eye array volume from 35% $V_{tot}$ to 12% $V_{tot}$, but has a negligible effect on an NS eye. Raising the tariff to $K_E=0.32$, reduces the simple eye’s array volume to just 3% $V_{tot}$ but in apposition models it is always > 30% $V_{tot}$. Despite this tenfold reduction in $\mathrm{\%}{V}_{tot}$ the total resources allocated to photoreceptor array increases from 35% to 57% (Figure 7d) because the relative contribution of photoreceptor energy consumption increases from 40% $C_{tot}$ to 55% $C_{tot}$. Finally, in simple eyes, the relative contribution of photoreceptor energy consumption is insensitive to $C_{tot}$, but in apposition models, it falls steadily with increasing $C_{tot}$, to around 5% $C_{tot}$ (Figure 7d). This difference between eye types can be attributed to differences in eye geometry (Figure 7a). As $C_{tot}$ increases, a simple eye’s photoreceptors maintain their dense packing, but an apposition eye’s photoreceptors become increasingly widely spaced as larger lenses force the boundaries between ommatidia further apart. We conclude that when resources are efficiently allocated, the differences in geometry that define eye type can profoundly influence both the impact of photoreceptor energy consumption on eye morphology and the distribution of costs within an eye.

In all optimised eye models, apposition and simple, the spatio-temporal information capacity, $H_{opt}$, increases sub-linearly with total investment, $C_{tot}$ (Figure 8a), making a bit of information more expensive in a larger eye. In simple eyes, $H_{opt}$ increases as ${\left(C_{tot}\right)}^{0.8}$ across the full range of $C_{tot}$ while in apposition models the exponent is lower, decreasing from 0.6 to 0.5 as $C_{tot}$ increases. Thus, the simple eye model is 10 times more efficient when $C_{\text{tot}}=1\times {10}^6$ μm³ sr⁻¹ and approximately 100 times more efficient when $C_{\text{tot}}=1\times {10}^{11}$ μm³ sr⁻¹ (Figure 8b). Increasing the photoreceptor energy tariff reduces the efficiency of all eye types, but this effect declines as apposition eye models increase in size (Figure 8b) because, for reasons of geometry, volume costs marginalise photoreceptor energy cost.

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This new measure of cost has several advantages. First, specific volume is defined by dimensions and angles that are measured anatomically and optically (Equations 1, 2, 5, and 6) and relates directly to benefits via optical, geometrical and physiological constraints on spatial acuity and sensitivity. Thus, like previous indicators of cost, namely eye radius, eye length, pupil area, and corneal area (Brandon et al., 2015; Brandon and Dudycha, 2014; Snyder et al., 1977; Thomas et al., 2020; Thomas et al., 2004), specific volume can evaluate investments and trade-offs that adapt eyes for specific purposes. Second, specific volume is more flexible and far-reaching than previous indicators of cost because it can be applied to eyes that are not well approximated by solid spheres. Third, it applies to small eye regions. Fourth, it provides a common currency for comparing investments in different eyes, eye regions, and components within an eye (Figures 5—7). This facility enables one to investigate how and why resources are distributed across visual fields and, as we demonstrate here, evaluate trade-offs between investments in an eye’s components. Given that biological processes and systems evolve to become more cost-efficient because costs depress fitness (Alexander, 1996; Niven and Laughlin, 2008; Taylor and Thomas, 2014), we suggest a fifth advantage: specific volume will help us understand how eyes are adapted and evolve.

How well does specific volume represent costs? Volume accurately measures one limiting resource, space, but because there are insufficient data, we have to make assumptions to calculate the costs of energy and materials. Assuming that the densities of an eye’s cells, extracellular materials and fluid-filled cavities are close to 1, space is a reasonable proxy for three costs that increase in proportion to mass: the energy expended carrying an eye (carriage cost), materials, and cellular energy consumption (Witter and Cuthill, 1993).

Equating mass and carriage cost is a reasonable starting point. Experiments on individual animals – a human walker carrying a backpack (Bastien et al., 2005), a flightless ant carrying nectar (Duncan and Lighton, 1994), and a flying beetle carrying a heavy weapon (Goyens et al., 2015) – show that the cost of carrying an additional load is the load’s mass times the individual’s mass specific metabolic rate when moving unloaded.

Turning to the costs of the materials and energy used within the eye, there is no data for an apposition eye’s dioptric apparatus. Nonetheless, its corneal lenses, pigment cells, and cone cells are densely packed with macromolecules, and because their optical function is passive (they refract and absorb), they contain very few, if any, mitochondria. Indeed, the corneal cuticle and the fluid-filled pseudo-cone of fly NS eye are extracellular (Hardie, 1985). Thus, the assumption that the costs of materials and energy for the dioptric apparatus increase in fixed proportion to volume is a reasonable starting point.

Cost estimates can be adjusted for situations in which costs per unit volume are not equal, as illustrated by our treatment of photoreceptor energy consumption. To support transduction, the photoreceptor array has an exceptionally high metabolic rate (Laughlin et al., 1998; Pangršič et al., 2005; Niven et al., 2007). We account for this higher energy cost by using the animal’s specific metabolic rate (power per unit mass and hence power per unit volume) to convert an array’s power consumption into an equivalent volume (Methods). Photoreceptor ion pumps are the major consumers of energy and the smaller contribution of pigmented glia (Coles, 1989) is included in our calculation of the energy tariff $K_E$ (Methods). The higher costs of materials and their turnover in the photoreceptor array can be added to the energy tariff $K_E$ but given the magnitude of the light-gated current (Laughlin et al., 1998), the relative increase will be very small. Thus, for our intents and purposes, the effects of these additional costs are covered by our models. For want of sufficient data, $K_E$ is uncertain; therefore, we model a reasonable range of values (Methods). Because the optimum eye configuration is most sensitive to $K_E$ in the lower half of this range (Figures 4 and 7), we are unlikely to seriously underestimate the effects of photoreceptor costs on eye morphology and efficiency.

The systematic increase in rhabdomere or rhabdom length, $L$, with spatial acuity observed in apposition eyes (Figures 1, 5 and 6) and simple eyes (see below), and predicted by our models (Figures 4 and 7), is an obvious manifestation of efficient resource allocation. Two studies of apposition eyes, one of dragonfly and the other of praying mantis (Labhart and Nilsson, 1995; Rossel, 1979), map the covariation of lens diameter, $D$, interommatidial angle, $\mathrm{\Delta }\varphi$, and rhabdom length, $L$, across an entire eye. Both show that when acuity is increased by enlarging the dioptric apparatus and increasing ommatidial density, that is by reducing $\mathrm{\Delta }\varphi$, $L$ is increased to improve $SN{R}_{ph}$. In both insects $L$ peaks in a fovea that is used to detect and capture prey, and in dragonfly dorsal fovea the $1.1\mathrm{m}\mathrm{m}$ rhabdomeres (the longest light sensitive waveguides reported in any eye, diurnal or nocturnal) are adapted to improve the detection and localisation of small flying prey (Labhart and Nilsson, 1995; Rigosi et al., 2017). However, specialisation for prey capture is unlikely to explain why $L$ varies systematically with acuity across an entire eye.

We find that the trend to increase rhabdomere length with acuity is more generally advantageous. The trend is widespread: looking across 28 eye regions drawn from 10 species of fast-flying insect, $L$ increases with acuity over a 25-fold range of interommatidial angle (Figures 5 and 6). Modelling shows that matching $L$ to acuity improves a general measure of performance, spatial information capacity and the widespread covariation of $L$ and acuity observed in apposition eyes follows the trend predicted by modelling (Figures 5 and 6). Finally, because information capacity captures the interplay between three basic determinants of image quality, $\mathrm{\Delta }\rho$, $\mathrm{\Delta }\varphi$, and $SN{R}_{ph}$, several tasks could benefit from an increase in $L$ with acuity.

Other factors can link $L$ to acuity. One possibility, developmental constraints impose inflexible scaling relationships on apposition eyes, can be dismissed. Scaling changes across apposition compound eyes (Perl and Niven, 2016; Scales and Butler, 2016; Taylor et al., 2019), and our empirical log–log plots of $L$ vs eye volume and $L$ vs $D$ continuously change slope (Figure 5d, e), as required for the efficient allocation of resources (Figure 4c). Head size is an obvious constraint: smaller insects have smaller heads; consequently, they will tend to have smaller eyes with lower acuity and shorter rhabdom(ere)s. The strongest evidence that lack of headroom reduces the length of photoreceptors comes from the fovea of a small insect, Holcocephala (Wardill et al., 2017), that is adapted for exceptionally high acuity (Table 2; Figure 5d). One must also consider the costs associated with enlarging the head by adding more eye, such as mechanical support, wind resistance, and conspicuousness. We suggest that more comparative studies are required to establish the influence of head size on eye design.

Then there is the difference in $L$ that is so obvious and widespread that it has been accepted without question – $L$ is generally much longer in diurnal apposition eyes than in diurnal simple eyes. Comparing eyes with rhabdomeric photoreceptors, $L$ ranges from 60 μm to 1100 μm in apposition eyes (Tables 2 and 3), and from 10 μm to 90 μm, in spiders’ simple eyes (Land, 1985). Efficient resource allocation provides an explanation: optimised simple eye models predict that $L$ is approximately x10 shorter than in apposition compound eyes of the same cost (Figure 7b). $L$ also increases with acuity (increasing $D$ and decreasing $\mathrm{\Delta }\varphi$) in optimised simple eye models and there is evidence that this matching occurs in diurnal spiders. $L$ increases with increasing $D$ when comparing eyes in different species (Land, 1985), different eyes in the same individual, and the same eye increasing in size and acuity as a juvenile spider grows to adulthood (Goté et al., 2019). However, more accurate models of these spiders’ eyes are needed to prove that this covariation is increasing efficiency. The degree to which the lengthening of cone outer segments in primate fovea (Provis et al., 2013) increases the efficiency of resource utilisation is also an open question. A critical ingredient, a physiological constraint that links investments in outer segments to noise levels in full daylight, has yet to be identified in ciliary photoreceptors. In addition, the economics of vertebrate simple eyes differs from invertebrate simple eyes in two ways. First, a vertebrate outer segment is almost 10 times more effective at absorbing photons because its folded sheets of membrane pack visual pigment molecules more densely than arrays of tubes (microvilli) (Fain et al., 2010). Second, photoreceptors compete with neural circuits for resources within the eye (Kröger and Biehlmaier, 2009; Sterling and Laughlin, 2015).

In apposition eye models optimised for spatio-temporal information capacity, more than 40% of total investment, $C_{tot}$, is allocated to photoreceptor arrays (Figures 4e and 6e) and, in line with this theoretical finding, our analysis of data from diurnal insects shows that arrays occupy 30–75% of eye volume (Figures 5f and 6d). For simple eye models, the efficient allocation is 30–60% of $C_{tot}$ (Figure 7d). Our modelling identifies roles for two types of array cost, the rarely considered costs associated with array volume (Baden and Nilsson, 2022; Kirschfeld, 1976; Kröger and Biehlmaier, 2009), and the well-known energy costs associated with phototransduction (Ames, 2000; Fain and Sampath, 2021; Laughlin et al., 1998; Okawa et al., 2008; Pangršič et al., 2005). We find that this energy cost greatly influences the configuration of an efficient eye: increasing the energy tariff $K_E$ reduces array depth, $L$, as much as tenfold (Figures 4c—7b). Reducing $L$ reduces $SN{R}_{ph}$ (Equation 4), however, the $SN{R}_{ph}$ that remains is worth paying for because the %$C_{tot}$ allocated to photoreceptor energy consumption increases (Figures 4f and 7e). We conclude that because $SN{R}_{ph}$ is valuable, photoreceptor costs are always significant and this necessitates efficient resource allocation. It follows that photoreceptor costs influence the morphologies of both photoreceptor array and dioptric apparatus because optics and array compete for the resources invested in an eye.

Photoreceptor energy cost has more impact on both the total cost and the design of a simple eye than an equivalent apposition compound eye because the simple eye has more photoreceptors per unit volume (Figure 7a). Applying our lowest energy tariff, $K_E=0.04$ μm³ per microvillus, to an optimised apposition eye model has almost no effect on the depth of the photoreceptor array, $L$, but in the optimised simple eye model, it reduces $L$ by more than 80%, and array volume by more than 60% (Figure 7c). Despite this shrinkage, investment in the simple eye model’s photoreceptor array increases from 35% $C_{tot}$ to 50% $C_{tot}$ because energy costs have risen from 30% $C_{tot}$ to 55% $C_{tot}$ (Figure 7d). Also, in simple eye models, the percentages of $C_{tot}$ allocated to the photoreceptor array volume and energy change very little as $C_{tot}$ increases, whereas in apposition eyes the relative contribution of photoreceptor energy consumption falls steeply as photoreceptors become increasingly widely spaced (Figure 7a). Thus, the distribution of costs within an efficiently configured eye changes with eye type, according to the eye’s typical geometry.

In both apposition and simple eye models, efficiency decreases with increasing $C_{tot}$ (Figure 8b), making a bit of information more expensive in a larger eye. The decline in efficiency with increasing eye size is steeper in apposition models, as expected from the well-known scaling relationships enforced by lens diffraction and eye geometry: spatial resolution increases as eye radius, $R$, in a simple eye and as $\sqrt{R}$ in an apposition eye (Kirschfeld, 1976). However, using $R$ as a measure of investment fails to identify the contribution of the photoreceptor array and fails to account for the fact that an apposition eye is not a solid sphere. When our more realistic measure of cost, specific volume, is used, a Drosophila size apposition eye is 10 times less efficient than a simple eye of the same cost and a dragonfly size eye is 100 times less efficient (Figure 8b). In both simple and apposition eyes, efficiency is reduced by increasing the photoreceptor energy tariff $K_E$. This effect is much greater in simple eyes (Figure 8b); thus, as also found for the reduction of photoreceptor length (Figure 7b), $K_E$ has more impact on the design of simple eyes.

We build on theoretical studies that show how constraints imposed by apposition optics, eye geometry, and eye size determine spatial resolution and information capacity when eye radius $R$, and hence corneal area, represent the limiting resource (Howard and Snyder, 1983; Kirschfeld, 1976; Snyder et al., 1977). Howard and Snyder, 1983 introduced the physiological constraint, transduction unit saturation, and showed that it increases the optimum value of eye parameter, $p$, in bright light by placing a fixed ceiling on $SN{R}_{ph}$ that is equivalent to operating at a lower light level. We extended their approach by using a measure of cost, specific volume, that allows eye radius $R$ to vary while total cost remains constant. This added realism allows us to raise and lower the SNR ceiling imposed by transduction units by trading investments in rhabdomere length against investments in dioptric apparatus. In so doing, we discover that it is advantageous for photoreceptors to lengthen rhabdomeres so as to reduce shot noise. Indeed, in daylight, Calliphora vicina R1–6 achieve the lowest level of photoreceptor noise reported to date, an equivalent contrast of 0.0084 (Anderson and Laughlin, 2000). This extreme specialisation resembles Autrum’s proposition that apposition eye photoreceptors compensate for poor spatial resolution by increasing temporal resolution (Autrum, 1958), thereby achieving the highest flicker fusion frequencies known, in excess of 400 Hz (Tatler et al., 2000). Perhaps enhancing photoreceptor performance to compensate for optical inefficiency is a principle of apposition eye design?

Increasing temporal resolution increases photoreceptor energy consumption, so to improve efficiency, apposition eye photoreceptors match temporal bandwidth to the properties of the images they form (Niven et al., 2007; Laughlin and Weckström, 1993). Eye regions that have lower spatial resolution and encounter more slowly moving images save energy by reducing photoreceptor bandwidth (Burton et al., 2001). Although this matching makes large energy savings, it is not included in our models because, to reduce the burden of establishing proof of principle, we fix photoreceptor bandwidth at the high value measured in a fully light-adapted blowfly. Nonetheless, our models can accommodate the costs and benefits of changing temporal bandwidth because they operate in the frequency domain (Methods). Boosting temporal resolution will be more cost-effective in large apposition eyes because photoreceptor energy costs are marginalised by costs associated with array volume (Figure 4e, f), therefore investments of energy that increase temporal bandwidth are increasing the return on the much larger investment in array volume (e.g. carriage cost). Some fast-flying diurnal insects with large facet lenses reduce the carriage cost of the photoreceptor array by surrounding each ommatidium with a palisade of air-filled tracheae (Horridge, 1969; Laughlin and McGinness, 1978; Smith and Butler, 1991; Wardill et al., 2017). The volumes and savings involved can be large. Published micrographs of dragonfly dorsal eye (Horridge, 1969) suggest that air fills at least 50% of array cross section.

In principle, one can establish the efficiency of an insect’s investments in optics and photoreceptors by computing the performance surface for a model eye region that has the same parameters as the insect’s eye region, then observing how closely the insect’s eye region approaches the optimum configuration. We cannot do this at present because a critical parameter, the photoreceptor energy tariff $K_E$, has not been determined: the best we can do is use data from blowflies to estimate $K_E$’s likely range (Methods). Not knowing $K_E$ is a major impediment to obtaining exact fits of our models to data because our modelling has discovered that the morphology of an efficient eye is sensitive to photoreceptor energy costs: in apposition eyes, increasing $K_E$ across its likely range reduces the depth of the photoreceptor array by as much as 90% (Figure 4c). Nor can we use our models to infer $K_E$ by adjusting it to fit the data. Changing $K_E$ has very little effect on $D$ and $\mathrm{\Delta }\varphi$, therefore we must fit the model to $L$, and this requires several data points from eyes of the same species because $K_E$ will be species specific (Methods). In our most consistent data set, the NS eyes (Figure 5; Table 2), it is impossible to obtain a satisfactory fit: the 12 data points come from 7 species and there are never more than 2 points per species. There are several data points per species for fused rhabdom apposition eyes (Figure 6; Table 3), but in each of the four species, the data points are too widely scattered to obtain a useful fit. This scatter in values of $L$ is not surprising. Our generic fused rhabdom model glosses over the large differences in F-number and rhabdom diameter seen within the eye of each species (Table 3), and these differences probably reflect the specialisation of eye regions for different purposes, for example in the fovea, the detection, tracking, and interception of mates and prey, and in the periphery, the monitoring of optic flow and the detection of potential prey and predators (Menzel et al., 1991; van Hateren et al., 1989; Labhart and Nilsson, 1995; Straw et al., 2006). This proposition is supported by the observation that the empirical data for $D$ and $\mathrm{\Delta }\varphi$ (Figure 6a, b) are also more widely scattered than the data for the more homogeneous group, fly NS eyes (Figure 5a, b). Because our method calculates three basic determinants of the quality of the captured image, $\mathrm{\Delta }\varphi$, $\mathrm{\Delta }\rho$, and $SN{R}_{ph}$, it can be adapted to compute performance surfaces for these specialised tasks; however, the necessary theory has not been developed.

Having raised the question, efficient for what, we must examine our assumption that every eye region of a diurnal insect is adapted for vision in full daylight. The fact that insects in our data sets have larger lenses, interommatidial angles and eye parameters than our models predict (Figure 5) suggests that their eye regions are adapted for lower light levels (Snyder, 1979). We can think of two mutually inclusive reasons why a diurnal eye will not be adapted for vision in full daylight. The eye might be adapted for the time of day when the behaviour it supports is most profitable, for example insects that are most active early and/or late in the day to avoid predators and desiccation, and insects that prey on these crepuscular insects. Second, because an eye’s performance surface is flat-topped, an eye region optimised for an intermediate light intensity might, when averaged over the daylight hours, perform better than an eye adapted for bright light. Our models can be developed to investigate how costs and benefits influence the design of eyes adapted for lower light levels (see below), and modified to investigate two further reasons for increasing the eye parameter $p$, higher image speeds (Snyder, 1979) and increasing $D$ in ‘bright zones’ to improve the detection of small targets and small movements (van Hateren et al., 1989; Straw et al., 2006).

In summary, for want of both data and theory, we are currently unable to quantify the efficiency with which an apposition eye region performs the function for which it is adapted. Nonetheless, we have two reasons to conclude that flying diurnal insects allocate resources to the optics and photoreceptor arrays of their apposition eyes efficiently. First, the empirical values we extract from published data lie within or close to the boundaries defined by our theoretical curves for eyes that maximise a general-purpose measure of performance, information capacity (Figures 5 and 6). Second, the flat-topped performance surfaces have extensive high-efficiency zones within which an eye’s morphology can be changed while retaining >95% of the maximum information capacity (Figure 3). Thus, eyes that are not optimised to maximise information capacity can still be gathering information with high efficiency. This observation speaks to the value of deriving performance surfaces: one can consider competing demands, evaluate loss of function, and identify compromises.

Efficient resource allocation is observed at many levels of biological organisation, from molecules to life histories (Bigman and Levy, 2020; Garland et al., 2022; Nijhout and Emlen, 1998; Stearns, 1989). Eyes are no exception. In an apposition eye, optical radius, $R$, represents corneal surface area per unit solid angle, therefore models that maximise spatial acuity or information capacity at constant $R$ show that corneal area can be allocated to facet lenses to maximise spatial acuity and information capacity by trading spatial resolution for SNR (Snyder, 1977; Snyder et al., 1977). In the POL region of fly NS eye, a fixed resource, the number of microvilli in the central rhabdom, is allocated to a pair of photoreceptors, R7 and R8, to increase the efficiency of polarisation coding (Heras and Laughlin, 2017), and in primate eye, retinal area is allocated to cone spectral types to increase coding efficiency (Garrigan et al., 2010; Zhang et al., 2022).

Matching the sampling density of the photoreceptor array to lens resolving power (Nilsson and Land, 2012) also allocates the resources invested in photoreceptors more efficiently by avoiding wasteful overcapacity, according to the principle of symmorphosis (Piersma and Gils, 2011; Taylor and Weibel, 1981; Weibel, 2000). However, unlike the matching of $L$ to $\mathrm{\Delta }\varphi$ that allocates resources efficiently to optics and photoreceptors, the matching of capacities does not consider the individual cost–benefit functions of a system’s components. When these are taken into account, the allocation of investment that maximises efficiency can instal overcapacity in components with more favourable cost–benefit ratios, as demonstrated for bones in racehorse forelegs (Alexander, 1997) and chains of signalling molecules (Schreiber et al., 2002).

The method we introduce here can be developed to view several aspects of eye adaptation and evolution through a cost–benefit lens because it is flexible. Our approach is not wedded to information capacity: it supports other measures of visual performance because it relates investments to fundamental determinants of image quality. Moreover, the method can be applied to several trade-offs that influence eye design by relaxing the simplifying assumptions we make here. To give a concrete example, the trade-offs between acuity and sensitivity that adapt eyes for nocturnal or diurnal vision can be viewed through a cost–benefit lens by allowing F-number and rhabdom(ere) diameter to vary, and by adopting a method that calculates the mean and variance of photon transduction rate on $N_{vil}$ in rhabdomeres of any given length and variable cross section, at any biologically plausible light intensity (Heras and Laughlin, 2017).

One can also relax the restrictions we place on eye geometry. Our approach does not depend on spherical geometry; it depends on being able to relate the shapes, sizes and constituents of an eye’s components to their costs and performance. Advances in scanning, imaging and digital reconstruction offer opportunities to determine shapes, volumes and constituents directly, and advances in modelling optical and physiological processes are extending our ability to relate these properties to performance (Kim, 2014; Song et al., 2012; Sumner-Rooney et al., 2019; Taylor et al., 2019). These powerful new methods can take us closer to the goal of understanding how costs, benefits and innovative mechanisms drive eye evolution (Sumner-Rooney, 2018), and our flexible cost–benefit approach can support this endeavour.

Our approach also identifies factors that contribute to selective pressure and gauges their effects. For example, the pressure to reduce photoreceptor energy cost depends on a photoreceptor’s impact on an animal’s energy budget. This effect is captured by our photoreceptor energy tariff, $K_E$, because it relates a photoreceptor’s energy consumption to the animal’s metabolic rate (Methods). We also find that photoreceptor energy costs are marginalised when apposition eye size increases, and this encourages the use of wide bandwidth, high SNR photoreceptors with high metabolic rates. Most interestingly, the derivation of performance surfaces (Figure 3) reveals a large area, the robust high-performance zone within which an eye’s morphology can be adapted to a particular task while retaining more than 95% of a measure of general-purpose capability, information capacity. This low ‘loss of opportunity’ cost helps to explain the remarkable degree of fine tuning observed in apposition compound eyes (Burton et al., 2001; Land, 1997; Nilsson and Land, 2012; Taylor et al., 2019; Sumner-Rooney et al., 2019), in line with suggestions that robustness increases evolvability (Drack and Betz, 2022).

Two key steps in the early evolution of eyes were the stacking of photoreceptive membranes to absorb more photons, and the formation of optics to intercept more photons and concentrate them according to angle of incidence (Nilsson, 2013; Nilsson, 2021). Our modelling of well-developed image forming eyes shows that to improve performance, stacked membranes (rhabdomeres) compete with optics for the resources invested in an eye, and this competition profoundly influences both form and function. Thus, it is likely that competition between optics and photoreceptors was shaping eyes as lenses evolved to support low resolution spatial vision. If so, the developmental mechanisms that allocate resources within extant high-resolution eyes (Casares and McGregor, 2021), by controlling cell size and shape and, as our study emphasises, setting up gradients of shape and size across the eye, will have analogues or homologues in ancient eyes. Their discovery will not only tell us how Nature constructs perfectly contrived instruments, it will illuminate the evolutionary pathway to perfection.

Our calculations of the rate at which an eye’s photoreceptor array codes achromatic information, in bits per steradian per second, take account of five factors. (1) The statistics of the signals presented to an eye under natural conditions; (2) the blurring of the spatial image by the eye’s dioptric apparatus; (3) the sampling of the spatial image by the photoreceptor array; (4) the introduction of noise by the stochastic activation of a finite population of transduction units; and (5) the temporal smoothing of signals and noise during phototransduction. We calculate information rates in the frequency domain using methods developed for apposition eyes (de Ruyter van Steveninck and Laughlin, 1996; van Hateren, 1992; Howard and Snyder, 1983; Snyder et al., 1977).

Our generic simple eye model has the same F-number $F=2$ as our generic fly NS model, and its rhabdom diameter equals the rhabdomere diameter in fly $d_{rh}=1.9$ µm. The simple eye is hemispherical (Figure 7a), therefore the specific volume of optics is given by

and with a photoreceptor array with rhabdoms of length $L$, the total specific volume of the eye is given by

and the specific volume of the photoreceptor array by

The receptor spacing, $\mathrm{\Delta }\varphi$, depends on the lens focal length, $f$ and $d_{rh}$,

This is an important difference between simple eyes and apposition eyes. In apposition eyes, the model $\mathrm{\Delta }\varphi$ is can be varied at constant $f$ and $D$ by changing the optical radius $R$. However, in a simple eye with a diffraction limited lens, the relationship between the optical resolving power of the lens and the anatomical resolving power of the photoreceptor mosaic depends on F-number, $F$, as follows (Snyder, 1979). For a hexagonal mosaic of photoreceptors, the diffraction limit is reached when

When $\lambda$, the wavelength of light is $500\mathrm{n}\mathrm{m}$, and $d_{rh}=2$ µm, then $F=6.9$. It follows that our model simple eye with $F=2$ is undersampling by a factor of approximately 3.5. Running our simple eye model with $F=4$ had very little effect on the optimum $L$ (data not shown), suggesting that our conclusion that efficient simple eyes have much shorter rhabdoms than efficient apposition eyes is robust with respect to undersampling.

We estimate $v$, the number of microvilli per unit length of rhabdomere, for our NS eye model using findings from photoreceptors R1–6 in female blowflies. Dividing the number of microvilli in a rhabdomere, $6\times {10}^4$ (Howard et al., 1987), by L = 260 µm, which is towards the upper end of the range L = 220 to 280 µm reported for blowfly (Hardie, 1985), gives ν = 230 µm⁻¹. Although fly rhabdomeres vary in both diameter and degree of taper according to eye region, sex and species, and this sculpting adapts photoreceptor arrays to visual ecology, for example (Gonzalez-Bellido et al., 2011), we fix $v$ to simplify the derivation of points of principle.

For our generic model of a fused rhabdom apposition eye, we consider a column of six photoreceptors that, like the short visual fibres R1–6 in fly, terminate in the first optic neuropile. Their six rhabdomeres form the triradiate rhabdom described in locust (Wilson et al., 1978). This fused rhabdom has a cross-section that approximates an equilateral triangle in which each side is constructed by a pair of photoreceptors whose parallel microvilli are equivalent to a single fly R1–6 rhabdomere. Thus, each pixel is coded by a set of six rhabdomeres that are equivalent to three fly R1–6 rhabdomeres. Therefore, the number of transduction units coding a pixel is half that in fly and the resulting changes in signal and noise are simply accounted for by replacing the factor $(3\psi {)}^2$ in Equation 29 with ${\left(\frac{3\psi }{2}\right)}^2$, and $3\psi$ in Equations 33 and 34 with $\left(\frac{3\psi }{2}\right)$.

For our generic model of a diurnal simple eye, we assume that, as in spiders, the rhabdomeres of adjacent photoreceptors abut; that is there are no gaps between photoreceptors; therefore, a photoreceptor’s rhabdomeres project from a central column of cytoplasm. To compare with our apposition models, we assume that the diameter of the photoreceptor entrance aperture, $d_{rh}=1.9$ µm and contains microvilli projecting in three directions. In this case, the simple eye photoreceptor is equivalent to the triradiate fused rhabdom.