Nilsson, D.-E., and Pelger, S. ‘A pessimistic estimate of the time required for an eye to evolve.’ Proceedings of the Royal Society of London B 256 :53-58, 1994. 1Department of Zoology, Lund University, Helgonavagen 3, S-223 62 Lund, Sweden2Department of Genetics, Lund University, Solvegatan 29, S-223 62 Lund, Sweden

Received 3 December 1993: accepted 23 December 1993

Proc. R. Soc. Lond. B (1994) 256, 53-58 (c) 1994 The Royal Society

(Nilsson D-E. & Pelger S., "A pessimistic estimate of the time required for an eye to evolve," Proceedings of the Royal Society, London, B, 1994, Vol. 256, pp.53-58)

SUMMARYTheoretical considerations of eye design allow us to find routes along which the optical structures of eyes may have evolved.

If selection constantly favours an increase in the amount of detectable spatial information, a light-sensitive patch will gradually turn into a focused lens eye through continuous small improvements of design.

An upper limit for the number of generations required for the complete transformation can be calculated with a minimum of assumptions.

Even with a consistently pessimistic approach the time required becomes amazingly short: only a few hundred thousand years.

When Charles Darwin (1859) presented his theory of evolution he anticipated that the eye would become a favourite target for criticism. He openly admitted that the eye was by far the most serious threat to his theory, and he wrote: 'that the eye ... could be been formed by natural selection seems, I freely confess, absurd in the highest possible degree'. Although the problem is principally important, it gradually lost its scientific potency, and has now almost become a historical curiosity. But eye evolution continues to fascinate, although the question is now one of process rate rather than one of principle.

Estimates of the number of generations required to make a certain change to a simple quantitative character are easily made if the phenotypic variation, election intensity and heritability of the character are known (Falconer 1989).

The evolution of complex structures, however, involves modifications of a large number of separate quantitative characters, and in addition there may be discrete innovations and an unknown number of hidden but necessary phenotypic changes.

These complications seem effectively to prevent evolution rate estimates for entire organs and other complex structures.

An eye is unique in this respect because the structures necessary for image formations although there may be several, are all typically quantitative in their nature, and can be treated as local modifications of pre-existing tissues.

Taking a patch of pigmented light-sensitive epithelium as the starting point, we avoid the more inaccessible problem of photoreceptor cell evolution (Goldsmith 1990; Land & Fernald 1992).

Thus, if the objective is limited to finding the number of generations required for the evolution of an eye's optical geometry, then the problem becomes solvable.

We have made such calculations his outlining a plausible sequence of alterations leading from a light-sensitive spot all the way to a fully developed lens eve. The model sequence is made such that every part of it, no matter how small, results in an increase of the spatial information the eye can detect. The amount of morphological change required for the whole sequence is then used to calculate the number of generations required. Whenever plausible values had to be assumed, such as for selection intensity and phenotypic variation, we deliberately picked values that overestimate the number of generations.

Despite this consistently pessimistic approach, we arrive at only a few hundred thousand generations!

2. A MODEL OF EYE EVOLUTION

The first and most crucial task is to work out an evolutionary sequence which would be continuously driven by selection.

The sequence should be consistent with evidence from comparative anatomy, but preferably without being specific to any particular group of animals. Ideally we would like selection to work on a single function throughout the entire sequence.

Fortunately, spatial resolution, i.e. visual acuity, is just such a fundamental aspect and it provides the sole reason for an eye's optical design (Snyder et al. 1977; Nilsson 1990; Warrant & McIntyre 1993).

Spatial resolution requires that different photoreceptor cells have different fields of view (Snyder et al. 1977).

A comparison of their signals then gives information about the direction of the incident light. The smaller the field of view of each individual intensity channel, the better is the potential for accurate spatial information.

It does not matter if the spatial resolution is used for measuring self-motion, detection of small targets, or complicated pattern recognition, the fundamental aspect of information is the same, and so are

54 D.-E. Nilsson and S. Pelger The time required for an eye to evolve

Humans have been designing cameras for a century, and telescopes for several centuries. So, optics is well understood.

As we go down the eye sequence, we want visual acuity to be improving. Nilsson and Pelger measured the acuity by calculating spatial resolution .

(The basic idea is that each light sensitive cell really should have a slightly different field of view from any other cell. When that is true, then different cells receive different information. The whole eye gathers the maximum possible amount of information.)

Resolution is not the only issue. Things such as the size of the eye and the brightness of the scene also affect acuity. However, Nilsson and Pelger's equations take care of all this. For each step in the sequence, the equations say that the eye has improved. And, there is a smooth gradation between each step.

Specifically, Nilsson and Pelger broke the sequence down into 1% changes. What I mean by a 1% change is that a length (or whatever) got 1% larger or smaller. When you look at it that way, the whole sequence, from eye spot to fish eye, turns out to take 1829 1% changes.

Nilsson and Pelger have a graph which shows the logarithm of the resolution versus the number of 1% changes. The graph is a straight line. This means that there aren't any inefficient areas in the sequence, where a lot of change will cause only a little improvement. It also means that there aren't any areas in the sequence where change has no benefit, or has negative benefit.

The graph is proof that the sequence is, indeed, a Darwinian scenario.

( graphs and figure missing )

[y-axis] optical resolution / (cycles rad-1)

[x-axis] morphological change / units of original patch size

[y-axis] detectable resolution / (cycles rad-1)

[x-axis] aperture size / units of nodal distance

[y-axis] optical resolution / (cycles rad-1)

[x-axis] central refractive index

Figure 1. Strategies for improving spatial resolution in an evolving eye. (a) An originally flat light-sensitive patch, or retina, is gradually invaginated (solid line) to form a pit whose distal aperture keeps the size of the original patch. The optical resolution is calculated as the inverse of the field of view of a point in the centre of the retina. At various points on the curve, the deepening of the pit is interrupted and all morphological change is instead spent on constriction of the aperture (broken lines). Calculations are made for aperture constriction to start when the pit depth, P. is 0.1, 0.5, 1.0 and 1.5 times the original width of the patch. (b) Optimisation of lenses aperture. Continuations of the dashed P = 1 curve in (a), but with photon noise taken into account with equation (1). The three curves are calculated for ambient intensities (I) separated by two log units. The upper curve is thus for an intensity 10000 times higher than that for the lower curve. The intensity is in units normalised to the nodal distance (pit depth): photons per nodal distance squared per second per steradian. The unconventional use of nodal distance instead of micrometres in the unit allows the three curves to be interpreted as eyes differing in size by a factor of 10. assuming constant intensity, the upper curve is thus for an eye with h is 100 times larger than that for the lower curve.

There is an optimum aperture size, indicated vertical lines, beyond which resolution (maximum resolvable spatial frequency) cannot be improved without a lens. (c) The optical resolution plotted as a function of the gradual appearance of a graded-index lens. The spherical lens is assumed to fill the aperture, and to be 2.55 lens radii away from the retina (e.g. as in fish eyes (Fernald 1990). The central refractive index of the lens is plotted on the horizontal axis. From this value the refractive index is assumed to follow a smooth gradient down to 1.35 at the peripheral margin. The calculation demonstrates that optical resolution continuously improves prom no lens at all to the focused condition where the central refractive index is 1.59. The maximum resolution in the focused Condition will be limited by both photon noise, as in (b), and by diffraction in the lens aperture, but none of these limitations are significant within the range plotted. The vertical axis of all three graphs (a) (c) was made logarithmic to allow for comparisons of relative improvement. A doubling of performance is thus always given by the same vertical distance.

the demands on eye design (see Warrant & McIntyre 1993).

We let the evolutionary sequence start with a patch of light-sensitive cells, which is backed and surrounded by dark pigment, and we expose this structure to selection favouring spatial resolution. We assume that the patch is circular, and that selection does not alter the total width of the structures. The latter assumption is necessary to isolate the design changes from general alterations of the size of the organ. There are two ways by which spatial resolution can be gradually introduced: (i) by forming a central depression in the light sensitive patch; and (ii) by a constriction of the surrounding pigment epithelium Both these morphological changes reduce the angle through which the individual light-sensitive cells receive light. The relative effects that depression and constriction have on the eye's optical resolution is compared in figure 1a. Initially, deepening of the pit is by far the most efficient strategy, but when the pit depth equals the width (P= 1 in figure 1a), aperture constriction becomes more efficient than continued deepening of the pit. We would thus expect selection first to favour

depression and imagination of the light-sensitive patch, and then gradually change to favour constriction of the aperture During this process a pigmented-pit eye is first formed which continues gradually to turn into a pinhole eye (see Nilsson 1990).

As the aperture constricts, the optical image becomes increasingly well resolved, but constriction of the aperture also causes the image to become gradually dimmer, and hence noisier. It is the random. nature of photon capture that causes a statistical noise in the image. When the image intensity decreases, the photon noise increases in relative magnitude, and the low contrast of fine image details gradually drowns in the noise. If we assume that the retinal receptive field, pret and the optical blur spot, plens, are identical Gossans, with half-widths being the angle subtended by the' aperture at a central point in the retina (this effectively means that the retinal sampling density is assumed always to match the resolution of the optical image), then we can use the theory of Snyder (1979) and

The time required for an eye to evolve DA. Nilsson and S. Pelger 55 Nilsson and Pelger took the eye sequence and broke it down into 1% changes. What I mean by a 1% change is that some value (such as a length, or a protein concentration) got 1% larger or smaller. For example, if some value has to double, then the doubling is considered to be 70 steps. (Mathematically, 1.01 to the 70th is about equal to 2.) They concluded that the whole sequence, as shown, required 1829 steps, as follows:

http://www.don-lindsay-archive.org/creation/eye_time.html

http://www.don-lindsay-archive.org/creation/eye_stages.html

1.- The diagram below shows a simple eye spot. Let us assume it is in the skin of a multicellular creature. It has a dark backing, because that makes vision a bit more directional.

1.- 176 1% steps d = 1

Single Light Sensitive Cells in Multicellular Organisms

Lumbricus terrestris (the earthworm) has a single cell that is sensitive to light although there may be several such cells scattered around its body. The cell is close to the base of the epidermis (the skin) and is thus protected from harm but able to receive light.

Multicellular Simple Eye

Here we have the advantage of more light sensitive cells, enabling the organism to see dimmer lights. An example would be Stylaria Lacustris (acquatic annelid worm).

2.- Next, an inward dimple happens under the eye spot. The eye spot begins to be on the surface of a shallow pit or depression. This increases the visual acuity, and also protects the eye spot from damage.

2. - 362 1% steps d = 1.23

3.- The dimpling continues until the depth of the pit is about equal to its width. This is now much like the eye of a planarian (flatworm). 

 3.- 270 steps (1 %) d = 1.95

Cupulate Eye

The cells are now arranged in a depression or pit. This protects the light sensitive cells from harm but also gives enables the eye to give some sense of the direction from which the light is coming. An example of a simple cupulate eye is that found in the limpet (Patella). A more complex example is that of Nereis, a polychaete worm which has a cuticle overlaying the pit. This cuticle provides further protection for the light sensitive cells

4. Next, the rim of the pit begins to constrict. In camera terms, the eye begins to have an "aperture". At some point - perhaps now, or perhaps later - the pit fills with a clear jelly. This may be a small mutation, or it could just be that the creature is covered by a slime layer anyway. In either case, the jelly or slime helps to hold the shape of the pit, and helps to protect the light sensitive cells from chemical damage. And, the jelly keeps mud out.

 4.- 225 steps (1 %) d = 2.83

5.- The aperture continues to decrease. Visual acuity increases until the aperture gets so small that it begins to shut out too much light. There will come a point when the aperture is the perfect size. A bigger aperture gives worse eyesight, and a smaller one gives worse eyesight. (The exact size that is "perfect" depends on how bright the lighting is.) This is now much like the eye of a nautilus.

The formation of a dark chamber

The optics of this eye can be improved if the pit becomes deeper and forms a completely enclosed chamber with a pinhole pupil. The eye then acts as a pinhole camera and can form a clear image, rather than simply having a gross direction sense. As most light is now excluded from the pit, i.e. it is cast in deep shadow, a dark chamber has been formed. An example of such an eye can be found in the mollusc Nautilus.

5.-192 steps (1 %) d = 4.56

The vesicular eye

If an epithelium (skin) grows over the pinhole pupil of the above eye then the pupil is protected from blockage. Pigment between the light sensitive cells helps to prevent light scattering within the retina and hence improves the clarity of the image.

6.- The eye above (5.-) is a perfect "pinhole camera".

It can only be improved by adding a lens. To get a lens, one mutation is needed. The pit must be roofed over with a transparent layer. This mutation is not that strange. First, it could have happened at any time before this stage. (The original eye spot might have been covered.) Second, the transparent layer is useful, to keep a lensless eye from damage. And third, transparent materials are not hard to come by. (The human cornea is made from a protein which is also used elsewhere in the human body.) So, the next step is the transparent layer becoming a little thicker in the center. Suddenly it isn't just a layer. It is a lens.

  6.- 308 steps (1%) d = f = 3P

Image forming eyes in Cephalopods

A simple development from the above is the cuticle to thicken in the middle hence forming a lens. If the epithelium near the light sensitive cells also grows over and thickens in the middle we can have two lenses, an internal lens and an external lens (known as a cornea). Part of the cuticle can also act to block light - it forms an iris.

Note that although optically the eye of the squid, an invertebrate, is very similar to the eye of a mammal, a vertebrate, in the vertebrate retina light must travel through the retinal nerve fibres before striking the light sensitive cells. This is not the case for an invertebrate's retina.

Cat's use an interesting development in order to see in the dark. They have a tapetum which is a reflecting surface, positioned behind the retina. This acts to reflect light which would otherwise be absorbed in the eye back onto the light sensitive cells. This increase in light sensitivity is brought about with a loss in the clarity of the image formed.

7.- Now that the eye has a lens, the aperture is in the wrong place. The eye will be more acute if the lens moves inward, towards the center of curvature of the light-sensitive surface

7.- 296 steps (1 %) d = 4.73 f = 2P

8.- The lens continues to move inward. As it moves, the laws of optics say that a thicker and thicker lens is valuable.

Also, the refractive index of the center of the lens changes. This is possible because the lens is made from a mixture of proteins. The ratio of the proteins can be different in different places, so the lens material is not optically uniform. It is common for a biological lens to have a higher refractive index at the center than at the edges. This "graded index" is a very valuable property.

8.- d = 4.1 f = P

And we're done. This is a fish eye, complete with a spherical graded-index lens, placed at the exact center of the light-sensitive layer. The optical quality is excellent, being "aberration-free" over a 180 degree field of view.

Total: 1829 1% steps.

The compound eye

The most complex eye that has evolved is the compound eye found in arthropods (spiders, insects, crustaceans). This is similar to the image forming eye of the vertebrate in that there is a corneal lens in addition to a second lens. In a compound eye this second lens is known as a lens cylinder and the optics of the lens are quite different to a "normal" lens. Furthermore, instead of having a single image forming chamber the compound eye has many such chambers, each called an ommatidium. These ommatidia are arranged in a regular array to make up the compound eye. Although each ommatidium is capable of forming an image, there are only 7 light sensitive cells in each ommatidium and the field of view is very restricted. Consequently each ommatidium signals only the presence of light in a given direction. Because the ommatidia all point in slightly different directions an overall image of the world can be built up.

The presence of pigment cells between ommatidia helps to improve the resolution of the compound eye. At night some insects can withdraw the pigment separating the ommatidia which helps to increase the sensitivity of the eye to light at the expense of resolution.

The "TV" eye

An interesting aside in the evolution of eyes is the development of an eye that works in a similar way to a television camera. This eye belongs to Copilia Quadrata., a microscopic animal. A lens on the surface of Copilia forms an image deep within the body. A second lens is paired with a light sensitive cell and forms an image of this image. However, the lens/receptor pair is not static - it moves back and forth across the image formed by the first lens. This allows a single light sensitive cell to transmit information about objects extended in space. Over time an image of the world can be built up. In fact Copilia has two eyes, one on either side of the body, and they move outwards and inwards in synchrony at a rate from 0.5 up to 5 times a second. By comparison a television camera signal repeats it's scanning 50 times a second.

http://www.cals.ncsu.edu/course/ent425/tutorial/photo.html

PHOTORECEPTORS

Compound EyesA pair of compound eyes are the principle visual organs of most insects; they are found in nearly all adults and in many immatures of ametabolous and hemimetabolous orders.   As the name suggests, compound eyes are composed of many similar, closely-packed facets (called ommatidia) which are the structural and functional units of vision.   The number of ommatidia varies considerably from species to

species:   some worker ants have fewer than six while some dragonflies may have more than 25,000.

Externally,each ommatidium is marked by a convex thickening of transparent cuticle, the corneal lens.   Beneath the lens, there is often a crystalline cone secreted by a pair of semper cells.   Together, the lens and the crystalline cone form a dioptric apparatus that refracts incoming light down into a receptor region containing visual pigment.

The light-sensitive part of an ommatidium is called the rhabdom.   It is a rod-like structure, secreted by an array of 6-8 specialized neurons ( retinula cells), and centered on the optical axis just below the crystalline cone.   The rhabdom contains an array of closely packed microtubules where light-sensitive pigments (e.g. rhodopsin, etc.) are stored.   These pigments absorb certain wavelengths of incident light and generate nerve impulses through a photochemical process similar to that of vertebrates.

Most diurnal insects have pigment cells surrounding each ommatidium.   These cells limit a facet's field of view by absorbing light that enters through adjacent corneas.   Each facet points toward a slightly different part of the visual field -- in composite, they render a mosaic-like impression of the environment.   Nocturnal and crepuscular insects have pigment cells that do not completely isolate each facet.   Their ommatidia are stimulated by light from larger fields of view.   This produces a brighter but theoretically less distinct mosaic image.

Since insects cannot form a true (i.e. focused) image of the environment, their visual acuity is relatively poor compared to that of vertebrates.   On the other hand, their ability to sense movement, by tracking objects from ommatidium to ommatidium, is superior to most other animals.  

Temporal resolution of flicker is as high as as 200 images/second in some bees and flies (in humans, still images blur into constant motion at about 30 images/second).   They can detect polarization patterns in sunlight, and discriminate wavelengths in a range from ultraviolet to yellow (but not red).

More about Color Vision

Ocelli -- Simple eyesTwo types of "simple eyes" can be found in the class Insecta:   dorsal ocelli and lateral ocelli (=stemmata).   Although both types of ocelli are similar in structure, they are believed to have separate phylogenetic and embryological origins.

Dorsal Ocellus

Dorsal ocelli are commonly found in adults and in the immature stages (nymphs) of many hemimetabolous species.   They are not independent visual organs and never occur in species that lack compound eyes.   Whenever present, dorsal ocelli appear as two or three small, convex swellings on the dorsal or facial regions of the head.   They differ from compound eyes in having only a single corneal lens covering an array of several dozen rhabdom-like sensory rods.   These simple eyes do not form an image or perceive objects in the environment, but they are sensitive to a wide range of wavelengths, react to the polarization of light, and respond quickly to changes in light intensity.   No exact function has been clearly established, but many physiologists believe they act as an "iris mechanism" -- adjusting the sensitivity of the compound eyes to different levels of light intensity.

Lateral Ocellus

Lateral ocelli (=stemmata) are the sole visual organs of holometabolous larvae and certain adults (e.g. Collembola, Thysanura, Siphonaptera, and Strepsiptera).   Stemmata always occur laterally on the head, and vary in number from one to six on each side.   Structurally, they are similar to dorsal ocelli but often have a crystalline cone under the cornea and fewer sensory rods.   Larvae use these simple eyes to sense light intensity, detect outlines of nearby objects, and even track the movements of predators or prey.   Covering several ocelli on each side of the head seems to impair form vision, so the brain must be able to construct a coarse mosaic

of nearby objects from the visual fields of adjacent ocelli.

Extra-ocular PhotoreceptionSome (perhaps most) insects respond to changes in light intensity even when all known photoreceptive structures are rendered inoperative.   This dermal light sense has been attributed to the response of individual neurons in the brain and/or ventral nerve cord.   There is also convincing evidence that some insects can perceive infrared radiation (heat), although specific receptors for this ability have not been described.

Mechanoreceptors

Chemoreceptors

Return to ENT 425 HomePageReturn to Tutorial IndexLast Updated:   2 September 2001

John R. Meyer Department of Entomology NC State University

http://www.britannica.com/eb/print?tocId=65136

Photo reception in arthropods

Ocelli :

n, pl. The simple eyes of a spider, usually numbering eight. They may be arranged in two or more rows of two to four eyes. In some hunting spider families, like the salticids, http://mamba.bio.uci.edu/~pjbryant/biodiv/spiders/Lycosa%20sp2.htm

the foremost central eyes may be modified with a movable retina to achieve binocular vision and range estimation. All other ocelli may be simple motion or light detectors. Among spiders with eight eyes they are usually designated in pairs as the anterior medians, the anterior laterals, the posterior medians and the posterior laterals; with six in pairs they may be called the anteriors or the two pairs of laterals. The relative size, placement and color of spider eyes are important in identifying spiders.

Most spiders have eight eyes. Some have two, four, six, or even 12 eyes. Spiders’ eyes are singular and are called ocelli, unlike the compound eyes found on many insects. The main pair of eyes is always the spider’s middle pair, and this pair has a different construction than the lesser eyes. The main eyes are used for focusing on prey. The other eyes detect movement well, and allow the spider to have a very broad range of vision.

http://www.life.umd.edu/faculty/wilkinson/BSCI338/L9ltreception/sld001.htm

There is a science called population genetics, and it has mathematical formulae for how quickly favorable genetic changes can spread throughout a population of sexually reproducing creatures. From these formulae, Nilsson and Pelger concluded that the 1829 steps could happen in about 350,000 generations.

In real life, an eye could evolve a little more quickly than that, or more slowly. It would depend on how much the specific creatures were being pressured to change, and on whether vision was relevant to their lifestyle.

If one year equals one generation, then a fairly good eye could evolve in maybe a third of a million years. It is thought that animal life has been on earth for at least 600 million years. That is certainly enough time for eyes to have evolved many times over.

In fact, taxonomists say that eyes have evolved at least 40 different times, and and possibly as many as 65 times. There are 9 different optical principles that have been used in the design of eyes and all 9 are represented more than once in the animal kingdom.

Why so many? Well, because there was time.

Figure 2.

Representative stages of a model sequence of eye evolution. In the initial stage (1) the structure is a flat patch of light-sensitive cells sandwiched between a transparent protective layer and a layer of dark pigment. In stages 2 and 3 the photoreceptor and pigment layer (hereafter collectively termed the retina) invaginates to form a hemisphere. The protective layer deepens to form a vitreous body, which fills the cavity. The refractive index of the vitreous body is assumed to be 1.35, which is only slightly higher than that of water, and not enough to give the vitreous body any, significant optical effect. In stages 4 and 5 the retina continues to grow, but without changing its radius of curvature. This causes it gradual shift from deepening of the retinal pit to constriction of the distal aperture. The aperture size in stage 6 was chosen to reflect the typical proportions in real eyes of this type. In stages 6-8 a graded-index lens appears by a local increase in refractive index. The central refractive index of the lens grows from the initial value of 1.35 to 1.52 in the final stage. Simultaneously the lens changes shape from ellipsoid to spherical and moves to the centre of curvature of the retina. As the lens shrinks, it flat iris gradually forms by, stretching of' the original aperture. The focal length f) of the lens gradually shortens, and in stage 8 it equals the distance to the retina (P), producing it sharply focused system. The relative change in receptor diameter, required to keep sensitivity constant throughout the sequence, is indicated by the normalized receptor diameter d. The anatomical change between model stages is given as the number of 1 %, modification steps.

Warrant & McIntyre (1993) to obtain the maximum detectable spatial frequency, max as:

max = (0.375P/A) [ln (0.746A2 /I)]½ (1)

where A is the diameter of the aperture, P is the posterior nodal distance, or pit depth and I is the light intensity in normalized units of 'photons per nodal distance squared per second per steradian'. We can now use this relation to plot resolution against aperture diameter (figure

1b). For a given ambient intensity and eye size there is an optimum aperture size where noise and optical blur are balanced in the image. A large eye or high light intensity makes for an optimum aperture which is small compared with the nodal distance. When the aperture has reached the diameter which is optimal for the intensity at which the eye is used, there can be no further improvement of resolution unless a lens is introduced.

In a lensless eye, a distant point source is imaged as a blurred spot which has the size of the imaging aperture. A positive lens in the aperture will converge light such that the blur spot shrinks, without decreasing the brightness of the image. Most biological lenses are not optically homogeneous, as man-made lenses normally are (Fernald 1990; Nilsson 1990; Land & Fernald 1992). In fact, a smooth gradient of refractive index, like that in fish or cephalopod lenses, offers a superior design principle for making lenses: the optical system can be made more compact, and aberrations can be reduced considerably (Pumphrey 1961). A graded-index lens can be introduced gradually as a local increase of refractive index. As the focal length becomes shorter, the blur spot on the retina will become smaller. The effect this has on resolution was calculated by, using the theory of Fletcher et al. 1954) for an ideal graded-index lens (figure 1c). Even the weakest lens is better than no lens at all, so we call be confident that selection for increased resolution will favour such a development all the way. from no lens at all to a lens powerful enough to focus a sharp image on the retina (figure 1c).

Camera-type eyes of aquatic animals typically have a spherical graded-index lens which is placed in the centre of curvature of the retina (Land 1981 ; Land & Fernald 1992). With this arrangement they. achieve virtually, aberration-free imaging over a full 180° visual field. Another typical feature is that the focal length of the lens is 2.55 times the lens radius (Pumphrey 1961).

56 D.-E. Nilsson and S. Pelger The time required for an eye to evolve

This relation, called Mattiessen's ratio, represents the ideal solution for a graded-index lens with a central refractive index of 1.52 (Fletcher et al. 1954), a value close to the upper limit for biological material. However, the best position for a lens to be introduced in a pinhole eye is in the aperture, clearly distal to the centre of curvature of the retina. Because the central and peripheral parts of the retina will then be at different distances from the lens, there is no need for the lens to be spherical. In fact, an ellipsoid lens is better because it can compensate optically for the difference in retinal distance. Furthermore, the size of the first-appearing lens is determined by the aperture, and need not have the size which will finally be required. As the lens approaches focused conditions, selection pressure gradually appears to move it to the centre of curvature of the retina, to make it spherical, and to adjust its size to agree with Mattiessen's ratio.

Based on the principles outlined above, we made a model sequence of which representative stages are presented in figure 2. The starting point is a flat light-sensitive epithelium, which by invagination forms the retina of a pigmented pit eye. After constriction of the aperture and the gradual formation of a lens, the final product becomes a focused camera-type eye with the geometry typical for aquatic animals (e.g. fish and cephalopods).

The changes in size and position of the aperture cause variations in linage brightness in the model sequence. To account for this we have assumed that the receptor diameter is continuously modified such that the photon catch per receptor, a rid thus the signal to noise

ratio, is kept constant throughout the sequence. As the model is of arbitrary. size, we have used a normalized receptor diameter (d) which is 1 at stage 1 in figure 2.

3. QUANTIFICATION OF CHANGE

The model sequence of figure 2 contains a number of structural elements whose shape and size are gradually modified. To quantify these changes we calculate the number of sequential 1% steps of modification it takes between each stage in figure 2. For example, a doubling of the length of a structure takes 70 steps of 1% (1.0170 2). Note that the last step is twice as long as the first in this example. The principle of 1% steps can be applied to changes of any quantitative character. Each structure of the model eye was analysed individually, as if no change follows passively from any other.

There are unavoidable ambiguities in measuring morphological change, because a product will have to be compared with a subjectively chosen origin. It would thus be possible to claim that a doubling in length of a structure is really a three times stretching of the outer half. Both views are correct, but they give different quantifications of the change. Measurements of phenotypic variation in a population suffer from the same type of subjectiveness. As we are going to relate our measures of morphological change only to general estimates of phenotypic variation, we will be safe as long as we avoid unorthodox and strange ways of comparing origin and product. Our principles have been to use whole length measurements of straight structures, arc length of curved structures, and height and width of voluminous structures. Changes in radius of curvature were accounted for by calculating the arc length of both the distal and proximal surface of the structure. Refractive index was related to protein concentration, by assuming that values above 1.34 are due to proteins alone.

The calculated number of 1% changes required between each of the stages (see figure 2) was plotted against the optical performance of each stage, which was calculated as the number of resolvable image points within the eye's visual field (figure 3). The graph, shows that spatial resolution improves almost linearly with morphological change. There are thus no particularly inefficient parts of the sequence, when much change has to be made for little improvement of function.

Altogether 1829 steps of 1% are needed for entire model sequence. Natural selection would act simultaneously on all characters that positively affect the performance. In our model there are several transformations that would speed up the improvement of function if they occurred in parallel. True to our pessimistic approach, we deliberately ignored this and assumed that all 1829 steps of 1% change occur in series.

This is equivalent to a single structure becoming 1.01^1829 or 80129540 times longer. In terms of morphological modification, the evolution of an eye can thus be compared to the lengthening of a structure, say a finger, from a modest 10 cm to 8000 km, of a fifth of the Earth's circumference.

( Graph missing )

[y-axis] image points in visual field

[x-axis] transformation progress (1% steps)

Figure 3.

Potential optical performance of the model 4 sequence plotted against transformation progress. The, geometry of each stage in figure 2 sets an upper limit to the spatial resolution of the eye. The smallest resolvable detail is given by the solid angle of the field of view of a point in the centre of the retina. The number of resolvable picture elements that can be fitted into the hemispherical visual field of the eye gives a measure of the number of image points that the eye can maximally resolve. When this estimate of the eye's optical performance is plotted on a logarithmic scale against the transformation progress, it forms an almost straight line. The last stage of figure 2 is not included because its performance is strongly dependent on ambient light intensity and the absolute size of the eye. If the intensity is high and the eye large. the number of image points in the last stage may be up to six orders of magnitude larger than in the penultimate stage.

The time required for an eye to evolve D.-E. Nilsson and S. Pelger 57

4. THE NUMBER OF GENERATIONS REQUIRED

Having quantified the changes needed for a lens eye to evolve, we continue by estimating how many generations such a process would require. When selection acts on a quantitative character, a gradual increase or decrease of the mean value, m, will be obtained over the generations. The response, R, which is the observable change in each generation is given by the equation

R = h2i p or = h2iVm, (2)

where h2 is the heritability, i.e. the genetically determined proportion of the phenotypic variance, i is the intensity of selection, V is the coefficient of variation, which measures the ratio between the standard deviation, p , and the mean, m, in a population (Falconer 1989). For our estimate we have chosen h2 = 0.50, which is a common value for heritability, while deliberately low values were chosen for both i (0.01) and V (0.01) (see Lande 1980; Futuyma 1986; Barton & Turelli 1989; Falconer 1989; Smith 1989). The response obtained in each generation would then be R = 0.00005m, which means that the small variation and weak selection cause a change of only 0.005%, per generation. The number of generations, n, for the whole sequence is then given by 1.000 05n = 80 129 540, which implies that n = 363 992 generations would be sufficient for a lens eye to evolve by natural selection.

5. DISCUSSION

Eyes closely resembling every part of the model sequence can be found among animals existing today Salvini-Plawen & Mayr 1977). From comparative anatomy it is known that molluscs and annelids display a complete series of eye designs, from simple epidermal aggregations of photoreceptors to large and well-developed camera eyes (Salvini-Plawen & Mayr 1977; Land & Fernald 1992). The structural components of our model have counterparts with different embryological origin in different groups. For modelling purposes these differences are irrelevant because selection for the various functions will operate on whatever tissue is present at the place where the function is needed.

The development of a lens with a mathematically ideal distribution of refractive index may, at first glance seem miraculous. Yet the elevation of refractive index in the lenses of both vertebrates and cephalopods is caused by proteins that are identical or similar to proteins with other cellular functions (Doolittle 1988; Goldsmith 1990; Winstow & Kim 1991; Land & Fernald 1992). Selection has thus recruited gene products that were already there. Assuming that selection operates on small but random phenotypic variations, no distribution of refractive index is inaccessible to selection. It is an inevitable consequence of selection for improved resolution that the population average is continuously adjusted towards the ideal distribution of refractive index. The lens should thus be no more difficult to evolve than any other structure of the model.

It is important that the model sequence does not underestimate the amount of morphological change required. The only real threat to the usefulness of our model is that we may have failed to introduce structures that are necessary for a functional eye. Features of many advanced eyes, such as an adjustable iris and structures for distance accommodation, may in this context seem to be serious omissions from the model (2) sequence. The function of these structures is to make the eye more versatile so that it can perform maximally over a greater range of distances and ambient intensities. However, the improved function brought about by the sequence of modifications in our model does not in any way depend on the existence of these auxiliary structures. It is in fact the other way around: evolution of these refinements requires the existence of the structures that develop in our model

Vertebrates and cephalopods have a vascularized layer, the choroid, and a supporting capsule, the sclera, in their eyes (Walls 1942; Messenger 1981). The demand for blood supply and structural support comes from the general lifestyle and size of these animals, and the demands involve the entire body, not just the eyes. The complete absence of choroid and sclera in the well-developed camera eyes of polychaetes and gastropod molluscs Salvini-Plawen & Mayr 1977; Messenger 1981) shows that such structures are not mandatory for eye evolution, and thus are not needed in our model.

The organization and specialization of the cells in the retina require sonic attention. The photoreceptor cells of advanced eyes are certainly not identical to those of simple light-sensitive spots (Eakin 1963; Goldsmith 1990), but even primitive photoreceptors should be good enough to make improvements of optical resolution. Improvements of efficiency, and specializations for polarization sensitivity and colour vision. are in no way required for selection to favour improvements of spatial resolution. The nervous tissue in the vertebrate retina can also be ignored as this is clearly a part of the nervous system which just happens to reside in the eye. No invertebrate eyes have an arrangement of this kind (Salvini-Plawen & Mayr 1977).

As far as we can tell, no structure of the eye has been omitted whose presence or development could in any way impede the evolutionary process. Further, we can be sure that real selection would outperform our model sequence in finding the modifications that give the best improvement of function for the least morphological change.

Can we be sure that our calculations do not underestimate the number of generations required for the optical structures to evolve? Throughout the calculations we have used pessimistic assumptions and conservative estimates for the underlying parameters. Should one or perhaps even two of these assumptions or estimates in fact be optimistic, we can trust that the

remaining ones will at least compensate for the errors made. It is more likely, though, that the complete

58 D.-E. Nilsson and S. Pelger The time required for an eye to evolve

calculation substantially overestimates the number of generations required.

If we assume a generation time of one year, which is common for small and medium-sized aquatic animals, it would take less than 364 000 years for a camera eye to evolve from a light-sensitive patch. The first fossil evidence of animals with eyes dates back to the early Cambrian, roughly 550 Ma ago (Salvini-Plawen & Mayr 1977; Land & Fernald 1992). The time passed since then is enough for eyes to evolve more than 1500 times!

If advanced lens eyes can evolve so fast, why are there still so many examples of intermediate designs among recent animals? The answer is clearly related to a fact that we have deliberately ignored, namely that an eye makes little sense on its own. Although reasonably well-developed lens eyes are found even in jellyfish (Piatigorsky et al. 1989), one would expect most lens eyes to be useless to their bearers without advanced neural processing. For a sluggish worm to take full advantage of a pair of fish eyes, it would need a brain with large optic lobes. But that would not be enough, because the information from the optic lobes would need to be integrated in associative centres, fed to motor centres, and then relayed to the muscles of an advanced locomotory. system. In other words, the worm would need to become a fish. Additionally, the eyes and all other advanced features of an animal like a fish become useful only, after the whole ecological environment has evolved to a level where fast visually guided locomotion is beneficial.

Because eyes cannot evolve on their own, our calculations do not say. how long it actually. took for eyes to evolve in the various animal groups. However, the estimate demonstrates that eye evolution would be extremely fast if selection for eye geometry and optical structure is imposed the only limit. This implies that eyes can be expected to respond very rapidly to evolutionary changes in the lifestyle of a species. Such potentially rapid evolution suggests that the eye design of a species says little about its phylogenetic relationship, but much about its need for vision. It follows that the many primitive eye designs of recent animals may be perfectly adequate, and simply reflect the animal's present requirements. In this context it is obvious that the eye was never a real threat to Darwin's theory of evolution.

We are deeply grateful to Dr Eric Warrant, for his swift help with crucial parts of the resolution calculations. We also thank Professor Bengt O. Bengtsson, Professor Robert Hessler and Dr Eric Warrant for critically reading the manuscript. The work was supported by the Swedish Natural Science Research Council.

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