### How long does it take to evolve an eye?

Darwin famously anticipated that complex organs in general and eyes in particular would be raised as a serious objection to his theory of natural selection. In the Origin of Species, he wrote:

To suppose that the eye with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic aberration, could have been formed by natural selection, seems, I freely confess, absurd in the highest degree.

Of course, he did not think this objection posed any real threat to his theory, but acknowledged the prima facie difficulty of imagining how a complex organ like the eye could have evolved from a simple beginning through small gradual changes where each change would somehow improve the fitness of the organism.

Nilsson and Pelger (1994) outline just such a series of small gradual changes that would lead from a primitive light-sensitive structure to a full-fledged complex lens eye. Along the way, they also give a worst-case estimate of the time it would take for the entire sequence of changes to take place. The resulting estimate is surprisingly small.

Remember that we not only want to identify a sequence of small gradual changes that would lead from a relatively simple structure to a complex lens eye, but we also require that each gradual change somehow improve the fitness of the organism, that it be useful to it. What is the criterion that we use for determining whether a change improves fitness or not? Nilsson and Pelger (1994) assume that spatial resolution of the eye provides this criterion. In other words, in the scenario considered by Nilsson and Pelger (1994), there is a selective pressure for improved spatial resolution (or improved visual acuity).

The crucial figure is Figure 2 in their paper (see below). It identifies the key stages in their model of the evolution of the eye. The starting point is a flat sheet of light-sensitive cells or photo-receptors (1). Their focus is on the evolution of the optics of the eye, so they sidestep the issue of how the photo-receptors could have evolved. Without a lens, there are basically two ways the spatial resolution of the eye can be improved: by forming a central depression or a “pit” in the sheet of light-sensitive cells or by constricting the aperture. It turns out that, initially, forming a pit and deepening it improves spatial resolution more than constricting the aperture, so that’s the route natural selection chooses initially (2-3). When the depth of the pit equals its width (3), constricting the aperture becomes more advantageous than deepening the pit further. So, that’s what natural selection does next (4-5): constrict the aperture. There is an optimal aperture size beyond which constricting the aperture further worsens the spatial resolution rather than improve it (Equation 1). At that point (5), no improvement in spatial resolution is possible without introducing a lens. A graded-index lens starts to form in the aperture by a gradient increase in refractive index (6). The lens becomes more spherical and moves to the center of curvature (7-8), which improves spatial resolution. At the same time a flat iris forms by the stretching of the aperture (not quite sure why it has to be flat, does a non-flat iris cause spherical abberation?).

Okay, so how long does this all take? They first calculate the number of sequential %1 steps of changes required to go from the starting point to the end point. For example, any doubling of the length of a structure takes about 70 such steps (because $\Huge 1.01^{70} \approx 2$). They analyze each structure individually and add the number of steps required for each. This is unrealistic, because normally a change might affect multiple structures simultaneously. They claim this makes their estimate a worst-case estimate, but I’m not sure if this is strictly true: a change in one structure can increase the number of changes required in another structure, right? The end result is 1829 steps. This is approximately equivalent to starting with a unit length structure and making it into a structure of length 80129540 at the end!

More concretely, how many generations would it take to engender all this change? This can be calculated using a population genetic identity that relates the phenotypic change in a trait among a population ($R$) to the heritability of the trait ($h^2$), the intensity of selection ($i$) and the variation of the trait among the population:

$R = h^2 i Vm$

where $m$ is the mean value of the trait in the population and $V$ is the ratio of the standard deviation of the trait to the mean of the trait (so $Vm$ is the standard deviation of the trait among the population). This equation makes sense: intuitively phenotypic change would be expected to increase with higher heritability, stronger selection and larger variation in the population. Nilsson and Pelger (1994) choose conservative estimates for each of $h^2$, $i$ and $V$ (remember they want a worst-case estimate) and come up with $R = 0.00005m$, or in other words a %0.005 change per generation in the mean trait. Now plugging this into the equation for the total amount of change required for an eye to evolve, we get the number of generations, $n$, required for the eye to evolve:

$1.00005^n = 80129540$

or $n \approx 363992$. Assuming a lifetime of 1 year per generation, this is 363992 years! An amazingly short period of time!

Another interesting result is that the spatial resolution increases almost linearly in evolutionary time (Figure 3), which suggests that there is no bottleneck in the system where very small improvements in spatial resolution have to accumulate before large improvements can kick in (this would be a scenario that would worry Darwin).

An interesting question is: if it takes such a small amount of time to evolve an eye with superb spatial resolution, why are there so many intermediate designs which are clearly suboptimal for spatial resolution among recent animals? Nilsson and Pelger (1994) point out that eyes are useless on their own. It does not make sense to evolve a sophisticated eye with superb spatial resolution if the organism does not need to have it. This guy, for example, clearly doesn’t have any need for an eye, let alone an eye with high spatial resolution.