Labyrinth: Fitness landscapes as mazes, not mountains

Tonight, I am passing through Toulouse on my way to Montpellier for the 2nd Joint Congress on Evolutionary Biology. If you are also attending then find me on 21 August at poster P-0861 on level 2 to learn about computational complexity as an ultimate constraint on evolution.

During the flight over, I was thinking about fitness landscapes. Unsurprising — I know. A particular point that I try to make about fitness landscapes in my work is that we should imagine them as mazes, not as mountain ranges. Recently, Raoul Wadhwa reminded me that I haven’t written about the maze metaphor on the blog. So now is a good time to write on labyrinths.

On page 356 of The roles of mutation, inbreeding, crossbreeding, and selection in evolution, Sewall Wright tells us that evolution proceeds on a fitness landscape. We are to imagine these landscapes as mountain ranges, and natural selection as a walk uphill. What follows — signed by Dr. Jorge Lednem Beagle, former navigator of the fitness maze — throws unexpected light on this perspective. The first two pages of the record are missing.

… The universe (which others call the Fitness Landscape) is composed of an indefinite number of hexagonal galleries. In the center of each gallery is a number, bounded by a low railing. The arrangement of the galleries is always the same: 72 doors, 12 to each side, line each of the hexagonal’s six sides; the height of the doors, floor to ceiling, is hardly greater than the height of a normal navigator. Each door opens to another gallery, identical to the first — identical in fact to all. Some doors, you pull open by a big vertical handle. Others, your push open via round push-plate. Each features a number. Light is provided by certain spherical fruits that bear the name “bulbs.” There are two of these bulbs in each hexagon, set crosswise. The light they give is insufficient, and unceasing.

The only difference between galleries is the numbers on the doors and center, and which doors are push or pull. The former follow no pattern that I could discern, but the latter have regularity.

Some five hundred years ago, this regularity of doors allowed a navigator of genius to discover the fundamental law of the Fitness Landscape. This philosopher observed that when you move between two galleries, all doors are oriented the same way except the door between them — that door, of course, is push for one gallery and pull for the other. Further, the number of each side of the door corresponds to the number on the floor of the gallery to which it leads. She further posited a fact which all navigators have since confirmed: In all the Fitness Landscape, there are no two galleries with an identical orientation of doors. From those uncontroversial premises, the navigator deduced that the Fitness Landscape is “total” and finite.

…

Like all the inhabitants of the Fitness Landscape, I travelled. Now that my eyes can hardly make out what I myself have written, I am preparing to die, a few leagues from the hexagon where I was born. But I wanted to leave a record.

My first gallery was numbered 890,528. This is a large number and it made me happy. But when I noticed that some doors in my gallery had numbers larger yet, I began to feel anxious. I tried pushing or pulling open many of the doors but realized that I am not strong enough to open doors with numbers lower than the gallery that I was in, but ones with larger numbers do give way. In fact, I found that the higher its number, the easier the door is to open for me. But this is not the case for all the navigators I met in my travels.

To relieve my anxiety, I left my first gallery. I chose a door inscribed with 890,604. I walked quickly straight across that hexagon to door 890,612 and then straight across again to 891,109. However, I could not continue this path further. Upon crossed this third hexagon, I saw door 890,528. Since this door was 581 less than the gallery that I was in, I could not open it. But I knew that on the other side was birthplace.

There were still other doors in the third gallery, with numbers greater than 891,109, so eventually, I wandered further. To this day, I have always had a door to open. I suspect that my path is unbounded. Although, like the mystics, I can imagine one day entering a gallery with a number so large that no door exceeds it.

The great philosopher of past has guaranteed that such a gallery must exist. Unfortunately, she left no map to it.