Biological landscapes, surfaces, and morphospaces: what are they good for?

ammonite

Metaphors are rampant in both everyday language and in science, and while they are inevitable, readers of this blog also know by now that I’m rather skeptical of their widespread use, both in professional publications and, especially, when addressing the general public. (See here, here, here, and here.) One such problematic metaphor is that of so-called adaptive landscapes, or surfaces, in evolutionary biology, something on which I did a fair amount of research when I was running a laboratory of ecology and evolutionary biology.

My detailed criticism of the way the landscape metaphor has sometimes warped biologists’ thinking is detailed in a chapter that was published back in 2012 as part of a very interesting collection entitled The Adaptive Landscape in Evolutionary Biology, edited by Erik Svensson and Ryan Calsbeek for Oxford University Press. As it often happens, mine was the lone contribution from the token skeptic…

Few metaphors in biology are more enduring than the idea of adaptive landscapes, originally proposed by Sewall Wright in 1932 as a way to visually present to an audience of typically non-mathematically savvy biologists his ideas about the relative role of natural selection and genetic drift in the course of evolution. The metaphor was born troubled, not the least reason for which is the fact that Wright presented different diagrams in his original paper that simply cannot refer to the same concept and are therefore hard to reconcile with each other. For instance, in some usages, the landscape’s non-fitness axes represent combinations of individual genotypes, while in other usages the points on the diagram represent gene or genotypic frequencies, and so are actually populations, not individuals.

typical (hypothetical) fitness landscape

Things got even more confusing after the landscape metaphor began to play an extended role within the Modern Synthesis in evolutionary biology and was appropriated by G.G. Simpson to further his project of reconciling macro- and micro-evolution, i.e. to reduce paleontology to population genetics. This time the non-fitness axes of the landscape were phenotypic traits, not genetic measures at all. How one would then translate from one landscape to another (i.e., genes to morphologies) is entirely unaddressed in the literature, except for vague motions to an ill-defined and very rarely calculated “genotype-phenotype mapping function.”

These are serious issues, if we wish to use the landscape metaphor as a unified key to an integrated treatment of genotypic and phenotypic evolution (as well as of micro- and macro-evolution). Without such unification evolutionary biology would be left in the awkward position of having two separate theories, one about genetic change, the other about phenotypic change, and no conceptual bridge to connect them.

To try to clarify things a bit, I went through the available literature and arrived at a typology of four different kinds of “landscapes” routinely used by biologists:

Fitness landscapes. These are the sort of entities originally introduced by Wright. The non-fitness dimensions are measures of genotypic diversity. The points on the landscape are typically population means, and the mathematical approach is rooted in population genetics. (see figure above)

Adaptive Landscapes. These are the non straightforward “generalizations” of fitness landscapes introduced by Simpson, where the non-fitness dimensions now are phenotypic traits. The points on the landscape are populations speciating in response to ecological pressures or even above-species level lineages (i.e., this is about macro-evolution). There is — with very special exceptions discussed in my paper — no known way to move from fitness to adaptive landscapes or vice versa, even though this is usually assumed by authors.

Fitness surfaces.These were introduced by Russell Lande and Steve Arnold back in the ‘80s to quantify the study of natural selection. Here phenotypic traits are plotted against a surrogate measure of fitness, and the landscapes are statistical estimates used in quantitative genetic modeling. The points on the landscape can be either individuals within a population or population means, in both cases belonging to a single species (i.e. this is about micro-evolution).

Morphospaces. These were first articulated by paleontologist David Raup in the mid-’60s, and differ dramatically from the other types for two reasons: (a) they do not have a fitness axis; and (b) their dimensions, while representing phenotypic (“morphological”) traits, are generated via a priori geometrical or mathematical models, i.e. they are not the result of observational measurements. They typically refer to across species (macro-evolutionary) differences, though they can be used for within-species work as well.

The first thing to note is that there are few actual biological examples of fitness landscapes (Wright-style) or Adaptive Landscapes (Simpson-style) available, while there is a good number of well understood examples of morphospaces (Raup-style) and particularly of adaptive surfaces (Lande–Arnold style). These differences are highly significant for my discussion of the metaphor. The paper summarizes examples — both conceptual and empirical — of each type of landscape and the complex, often barely sketched out, relationships among the different types.

When it comes to asking what the metaphor of landscapes in biology is for, we need to distinguish between the visual metaphor, which is necessarily low-dimensional, and the general idea that evolution takes place in some sort of hyper-dimensional space. Remember that Wright introduced the metaphor because his advisor suggested that a biological audience at a conference would be more receptive toward diagrams than toward a series of equations. But of course the diagrams are simply not necessary for the equations to do their work. More to the point, subsequent research by my former University of Tennessee colleague Sergey Gavrilets and his collaborators has shown in a rather dramatic fashion that the original (mathematical) models were far too simple and that the accompanying visual metaphor is therefore not just incomplete, but highly misleading. It turns out that hyper-dimensional dynamics are very much qualitatively different from the low-dimensional ones originally considered by Wright.

In a very important sense Wright’s metaphor of fitness landscapes was meant to have purely heuristic value, to aid biologists to think in general terms about how evolution takes place, not to actually provide a rigorous analysis of, or predictions about, the evolutionary process (it was left to the math to do that work). Seen from this perspective, fitness landscapes have been problematic for decades, generating research aimed at solving problems — like the so-called peak shift one (how do populations stuck on a local fitness peak “shift” to a higher one?) that do not actually exist as formulated, since high-dimensional landscapes don’t have “peaks” at all, as their topology is radically different.

There are problems also with the Lande-Arnold type landscapes (discussed in the paper), but here I want to shift to some good news: the actual usefulness of the fourth type of landscape: Raup-style morphospaces. One of the best examples was produced by Raup himself, with crucial follow-up by one of his graduate students, John Chamberlain. It is a study of potential ammonoid forms that puts the actual (i.e., not just heuristic) usefulness of morphospaces in stark contrast with the cases of fitness and adaptive landscapes. Ammonoids, of course, were beautiful shelled marine invertebrates that existed in a bewildering variety of forms for a good chunk of Earth’s biological history, and eventually went extinct 65 million years ago, together with the dinosaurs. This is going to be a bit technical, but stick with me, it will be worth it.

Raup explored a mathematical-geometrical space of ammonoid forms defined by two variables: W, the rate of expansion of the whorl of the shell; and D, the distance between the aperture of the shell and the coiling axis. Raup arrived at two simple equations that can be used to generate pretty much any shell morphology that could potentially count as “ammonoid-like,” including shells that — as far as we know — have never actually evolved in any ammonoid lineage. Raup then moved from theory to empirical data by plotting the frequency distribution of 405 actual ammonoid species in W/D space and immediately discovered two interesting things: first, the distribution had an obvious peak around 0.3 <D <0.4 and W near 2. Remember that this kind of peak is not a direct measure of fitness or adaptation, it is simply a reflection of the frequency of occurrence of certain forms rather than others. Second, the entire distribution of ammonoid forms was bounded by the W = 1/D hyperbola, meaning that few if any species crossed that boundary on the morphospace. The reason for this was immediately obvious: the 1/D line represents the limit in morphospace where whorls still overlap with one another. This means that for some reason very few ammonites ever evolved shells in which the whorls did not touch or overlap.

one-peak ammonoid morphospace

Raup’s initial findings were intriguing, but they were lacking a sustained functional analysis that would account for the actual distribution of forms in W/D space. Why one peak, and why located around those particular coordinates? Here is where things become interesting and the morphospace metaphor delivers much more than just heuristic value. John Chamberlain, a student of Raup, carried out experimental work to estimate the drag coefficient of the different types of ammonoid shells. His first result clarified why most actual species of ammonoids are found below the W=1/D hyperbola: shells with whorl overlap have a significantly lower drag coefficient, resulting in more efficiently swimming animals.

However, Chamberlain also found something more intriguing: the experimental data suggested that there should be two regions of the W/D morphospace corresponding to shells with maximum swimming efficiency, while Raup’s original frequency morphospace detected only one peak. It seemed that for some reason natural selection found one peak, but not the other. Four decades had to pass from Raup’s paper for the mystery of the second peak to be cleared up: the addition of 597 new species of ammonoids to the original database showed that indeed the second peak had also been occupied!, a rather spectacular case of confirmed prediction in evolutionary biology, not exactly a common occurrence, particularly in paleontology.

two-peak ammonoid morphospace, with representative shell forms

So, is the landscape metaphor in biology useful? It depends. The original versions, those introduced by Sewall Wright to make his math accessible to his colleagues, have been highly influential for decades, and yet have arguably channeled both empirical and theoretical research in unproductive directions, inventing problems (like the peak shift one) that arguably do not exist, at least not as formulated. The Lande-Arnold landscapes, which I have not discussed in this post, but do treat in the paper, have a mixed record. They have been heuristically useful for biologists interesting in quantifying natural selection in the field, but have also arguably brought about a degree of tunnel vision in both the theoretical and empirical study of that most important concept in modern evolutionary theory. Morphospaces, by contrast, have a very good record of being useful in terms of generating insight into the evolution of animal (and plant) form, and yet, they are actually the least commonly deployed version of the landscape idea in the technical literature. And because population genetics, with its mathematical approach, is considered more sophisticated than paleontology, things are unlikely to change in the near future. Unfortunately.