# Nearly Neutral Networks and Holey Adaptive Landscapes

My holiday reading on modularity has led me into some of the literature on the evolution of complexity.  Some of the most interesting work in theoretical biology that I’ve read in a while relates to the ideas of nearly neutral networks and holey adaptive landscapes, an area developed by Sergey Gavrilets at the University of Tennessee.  The various papers of his to which I refer can be accessed on his website.  I find his explanations very clear, but recognize that this work is fairly technical stuff and my understanding of it is greatly facilitated by previous experience with similar models in the context of epidemics on networks (believe it or not). Nonetheless, a reasonably accessible introduction can be found in his 2003 chapter, “Evolution and speciation in a hyperspace: the roles of neutrality, selection, mutation and random drift.” I have based much of my discussion here on this paper along with his 1997 paper in JTB.

The father of American population genetics and Modern Evolutionary Synthesis pioneer Sewall Wright first came up with the metaphor of the adaptive landscape in 1932.  The basic idea is a kind of topographic map where the map coordinates are given by the genotype and the heights above these coordinates are given by the fitnesses associated with particular genotype combinations.  A landscape, of course, is a three dimensional object.  It has a length, a width (or latitude and longitude) and height.  This particular dimensionality turns out to be very important for this story.

A major concern that arises from the idea of an adaptive landscape is how populations get from one peak to another.  In order to do this, they need to pass through a valley of low fitness and this runs rather counter to our intuitions of the way natural selection works.  The usual way around this apparent paradox is to postulate that populations are small and that random genetic drift (which will be more severely felt in small populations) moves the population away from its optimal point on the landscape.  Once perturbed down into a valley by random forces, there is the possibility that the population can climb some other adaptive peak.

This is a slightly unsatisfying explanation though.  Say that we have a finite population of a diploid organism characterized by a single diallelic locus. Finite populations are subject to random drift. The size of the population is $N$.  Assume that the fitnesses are $w_{AA}=1$, $w_{Aa}=1-s$, and $w_{aa}=1$. This is actually a very simple one-dimensional adaptive landscape with peaks at the ends of the segment and a valley in between.  Assume that the resident population is all $AA$.  What happens to a mutant $a$ allele? We know from population genetics theory that the probability that a completely neutral (i.e., $s=0$) mutant allele reaching fixation is $1/2N$.  Russ Lande has shown that when the $s>0$ this probability becomes:

where $erf()$ is the error function $erf(t) = 2/\sqrt{\pi} \int_0^t e^{-y^2} dy$.

When $Ns=20$ (say a population size of 200 and a fitness penalty of $s=0.1$), this probability is approximately $U=10^{-8}$.  So for quite modest population size and fitness disadvantage for the heterozygote, the probability that the population will drift from $AA$ to $aa$ is very small.  This would seem to spell trouble for the adaptive landscape model.

Gavrilets solved this conundrum — that moving between peaks on the adaptive landscape appears to require the repeated traversing of very low-probability events — apparently by thinking a little harder about the Wrightean metaphor than the rest of us.  Our brains can visualize things very well in three dimensions.  Above that, we lose that ability.  Despite all the Punnett squares we may have done in high school biology, real genotypes, of course, are not 2 or 3 dimensional.  Instead, even the simplest organism has a genotype space defined by thousands of dimensions.  What does a thousand dimensional landscape look like? I haven’t the faintest idea and I seriously doubt anyone else does either.  Really, all our intuitions about the geometry of such spaces very rapidly disappear when we go beyond three dimensions.

Using percolation theory from condensed matter physics, Gavrilets reveals a highly counter-intuitive feature of such spaces’ topology. In particular, there are paths through this space that are very nearly neutral with respect to fitness.  This path is what is termed a “nearly neutral network.” This means that a population can drift around genotype space moving closer to particular configurations (of different fitness) while nonetheless maintaining the same fitness.  It seems that the apparent problem of getting from one fitness peak to another in the adaptive landscape is actually an artifact of using low-dimensional models. In high-dimensional space, it turns out there are short-cuts between fitness peaks.  Fitness wormholes?  Maybe.

Gavrilets and Gravner (1997) provide an example of a nearly neutral network with a very simple example motivated by Dobzhansky (1937).  This model makes it easier to imagine what they mean by nearly neutral networks in more realistic genotype spaces.

Assume that fitness takes on one of two binary values: viable and non-viable. This assumption turns our space into a particularly easy type of structure with which to work (and it turns out, it is easy to relax this assumption).  Say that we have a three diallelic loci ($A$, $B$, and $C$).  Say also that we have reproductively isolated “species” whenever there is a difference of two homozygous loci — i.e., in order to be a species a genotype must differ from the others by at least homozygous loci.  The reproductive isolation that defines these species is enforced by the fact that individuals heterozygous at more than one locus are non-viable. While it may be a little hard to think of this as a “landscape”, it is.  The species nodes on the cube are the peaks of the landscape.  The valleys that separate them are the non-viable nodes on the cube face.  Our model for this is a cube depicted in this figure.

Now using only the visible faces of our projected cube, I highlight the different species in blue.

The cool thing about this landscape is that there are actually ridges that connect our peaks and it is along these ridges that evolution can proceed without us needing to postulate special conditions like small population size, etc. The paths between species are represented by the purple nodes of the cube.  All the nodes that remain black are non-viable so that an evolutionary sequence leading from one species to another can not pass through them.  We can see that there is a modest path that leads from one species to another — i.e., from peak to peak of the fitness landscape. Note that we can not traverse the faces (representing heterozygotes for two loci) but have to stick to edges of the cube — the ridges of our fitness landscape.  There are 27 nodes on our cube and it turns out that 11 of them are viable (though the figure only shows the ones visible in our 2d projection of the cube).

So much for a three-dimensional genotype space. This is where the percolation theory comes in. Gavrilets and Gravner (1997) show that as we increase the dimensionality, the probability that we get a large path connecting different genotypes with identical fitness increases.  Say that the assignment of fitness associated with a genotype is random with probability $p$ that the genotype is viable and $1-p$ that it is non-viable.  When $p$ is small, it means that the environment is particularly harsh and that very few genotype combinations are going to be viable. In general, we expect $p$ to be small since most random genotype combinations will be non-viable. Percolation theory shows that there are essentially two regimes in our space.  When $p, where $p_c$ is a critical threshold probability, the system is subcritical and we will have many small paths in the space.  When  $p>p_c$, we achieve criticality and a giant component forms, making a large viable evolutionary path  traversing many different genotypes in the space.  These super-critical landscapes are what Gavrilets calls “holey”. Returning to our three dimensional cube, imagine that it is a chunk of Swiss cheese.  If we were to slice a face off, there would be connected parts (i.e., the cheese) and holes.  If we were, say, ants trying to get across this slice of cheese, we would stick to the contiguous cheese and avoid the holes. As we increase the dimensionality of our cheese, the holes take up less and less of our slices (this might be pushing the metaphor too far, but hopefully it makes some sense).

A holey adaptive landscape holds a great deal of potential for evolutionary change via the fixation of single mutations.  From any given point in the genotype space, there are many possibilities for evolutionary change.  In contrast, when the system is sub-critical, there are typically only a couple of possible changes from any particular point in genotype space.

To get a sense for sub-critical and supercritical networks, I have simulated some random graphs (in the graph theoretic sense) using Carter Butts‘s sna package for R.  These are simple 1000-node Bernoulli graphs (i.e., there is a constant probability that two nodes in the graph will share an undirected edge connecting them).  In the first one, the probability that two nodes share an edge is below the critical threshold $p_c$.

We see that there are a variety of short paths throughout the graph space but that starting from any random point in the space, there are not a lot of viable options along which evolution can proceed. In contrast to the sub-critical case, the next figure shows a similar 1000-node Bernoulli graph with the tie probability above the critical threshold — the so-called “percolation threshold.”

Here we see the coalescence of a giant component.  For this particular simulated network, the giant component contains 59.4% of the graph.  In contrast, the largest connected component in the sub-critical graph contained 1% of the nodes.  The biological interpretation of this graph is that there are many viable pathways along which evolution can proceed from many different parts of the genotype space. Large portions of the space can be traversed without having to pass through valleys in the fitness landscape.

This work all relates to the concept of evolvability, discussed in the excellent (2008) essay by Massimo Pigliucci.  Holey adaptive landscapes make evolvability possible.  The ability to move genotypes around large stretches of the possible genotype space without having to repeatedly pull off highly improbable events means that adaptive evolution is not only possible, it is likely.  In an interesting twist, this new understanding of the evolutionary process provided by Gavrilets’s work increases the role of random genetic drift in adaptive evolution.  Drift pushes populations around along the neutral networks, placing them closer to alternative adaptive peaks that might be attainable with a shift in selection.

Super cool stuff.  Will it actually aid my research?  That’s another question altogether…

Another fun thing about this work is that this is essentially the same formalism that Mark Handcock and I used in our paper on epidemic thresholds in two-sex network models. I never cease being amazed at the utility of graph theory.

References

Dobzhansky, T. 1937. Genetics and the Origin of Species. New York: Columbia University Press.

Gavrilets, S. 2003. Evolution and speciation in a hyperspace: the roles of neutrality, selection, mutation and random drift. In Crutchfield, J. and P. Schuster (eds.) Towards a Comprehensive Dynamics of Evolution – Exploring the Interplay of Selection, Neutrality, Accident, and Function. Oxford University Press. pp.135-162.

Gavrilets, S., and J.Gravner. 1979. Percolation on the fitness hypercube and the evolution of reproductive isolation. Journal of Theoretical Biology 184: 51-64.

Lande, R.A. 1979. Effective Deme Sizes During Long-Term Evolution Estimated from Rates of Chromosomal Rearrangement. Evolution 33 (1):234-251.

Pigliucci, M. 2008. Is Evolvability Evolvable? Nature Genetics 9:75-82.

Wright, S. 1932. The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proceedings of the 6th International Congress of Genetics. 1: 356–366.

# On Modules

As the next installment in my series on evolution psychology (see previous posts here and here), I thought that I would write about some thoughts on evolutionary modules.  As should be obvious from previous posts, I have serious concerns about evolutionary psychology.  Nonetheless, I don’t want to repeat the knee-jerk criticisms that attended the rise of what you might call (and Symons (1989) did call) “Darwinian Anthropology.”  Like Anthropology more generally, I have found that the level of discourse in human evolutionary studies tends to be particularly low and this surely hinders progress toward our presumably shared goals of understanding human behavior, the origin and maintenance of human diversity, and how people respond to social, environmental, and economic changes.

In this spirit, I am taking seriously the idea of modularity.  The concept of “massive modularity” seems to be pretty central to just about any definition of modern EP and it is one of the ideas that I see as potentially most problematic.  A major question that naturally arises in the analysis of cognitive modularity is: what is a module?  There are two senses of modularity that you find discussed in the EP literature. For a good review of this, see Barrett and Kurzban (2006). In his highly influential (1983) book, Fodor popularized the concept of a cognitive module.  A Fodorian module is characterized by reflex-like encapsulation of critical functions.  It is thought to be anatomically localized, inaccessible to conscious thought and has shallow outputs.  Our senses and motor systems are examples of possible Fodorian modules, as are the systems that underlie language (Machery 2007).

In contrast to the Fodorian module is the second sense of modularity found in the EP literature, the evolutionary module. Like a Fodorian module, the evolutionary module is domain-specific or informationally encapsulated.  That is where the resemblance ends though.  Rather than being defined by a list of attributes, an evolutionary module is characterized by function.  An evolutionary module is a domain-specific cognitive mechanism that has been shaped by natural selection to perform a specific task.   There is no need here to specify their characteristic operating time, the shallowness of their outputs, or their anatomical localization.

Using engineering-inspired arguments about efficiency and design, the proponents of massive modularity suggest that the brain is really a collection of domain-specific modules.  These modules drive not just the reflex-like actions of our sensory-motor systems but also govern higher cognitive processes like reason, judgment, and decision-making.  The brain is not, as we typically conceive it, a single organ.  Rather it is a collection of special-purpose information processing organs.   Needless to say, such a position has been controversial.  Among the notable critics are Jerry Fodor himself, who wrote a whole book with the sarcastic title (referring to Steve Pinker’s (1997) book, How the Mind Works), The Mind Doesn’t Work That Way: The Scope and Limits of Computational Psychology.  Another notable critic is David Buller, the ostensible subject of my last two posts.

Barrett & Kurzban (2006) suggest that much of the controversy surrounding the EP concept of massive modularity arises from confusion over what is meant by a module in the EP sense.  That is, critics are thinking about Fodorian modules when the advocates of massive modularity have something entirely different in mind. Maybe.  I’m no expert, but the argument seems plausible for at least part of the controversy.   I have my own issues with modularity but I will save that for the paper that I am writing (and for which these posts serve as sketches to hopefully help me get some thoughts straight).

One point that I will make here is a fairly orthodox criticism of modularity.  In enumerating possible evolutionary modules, and noting that such modules require domain-specific input criteria, Barrett & Kurzban (2006: 630) include “systems specialized for making good food choices process only representations relevant to the nutritional value of different potential food items.”  Really? I’m not one to fall back on the weak “culture complicates things” argument, but I do think there are other things — including ones potentially important for fitness — involved in food choice than the nutritional quality of a potential foodstuff. Perhaps an anecdote is in order here.

A long time ago, my wife and I were taken out to a fancy Chinese restaurant in Kota Tua, Jakarta by a colleague who wanted to impress us with his esoteric knowledge of a variety of Asian cuisines.  He took the initiative and ordered for the table a range of items including tripe, jellyfish, pig trotters, and chicken feet. For a variety of complex social reasons, we felt it was in our interest to not seem like naïve rubes from America.  So, we ate everything unflinchingly and with smiles on our faces. These were not things we normally would have volunteered to eat (though we now regularly get jellyfish) but the social payoffs of eating these (at the time) unappealing items outweighed whatever distaste we may have experienced.

Clearly, this is a bit of a trivial example.  I nonetheless think that it highlights an extremely important aspect of human decision-making.  The optimal decision in a one dimensional problem may change when one increases the dimensionality of the problem, particularly when the elements of your (vector) optimand trade-off.  Sometimes the optimal nutritional choice is not the optimal choice with respect to social or cultural capital.  The person’s foraging decision is presumably one that balances the various dimensions of the problem. In a less trivial example, this is what Hawkes, O’Connell and Bird and Bird are suggesting is going on with some men’s foraging decisions  (summarized in this review by Bird & Smith (2005)).  According to their model, men make energetically suboptimal foraging decisions in order to signal their phenotypic quality to political allies and potential mates.  Food choice is thus a decision that balances the potential costs and benefits of at least three fitness-critical domains (energetics, politics, and reproduction).   The same logic can be applied to that other staple of EP, mate choice.  What people say they want on pen-and-paper surveys is not necessarily what they get when they actually choose a mate.  The problem is that one’s choice of mate spills over into so many other domains than simply future reproduction.  So it’s not simply a matter of the ideal mate being out of one’s league.  Sometimes, people actually prefer a mate who does not conform to their ideal physical type.

At the very least, this point seems to require positing the existence of yet another module that integrates the outputs of various lower-level modules.  Of course, this is beginning to sound more like a generalized reasoning process, the bane of EP.

There is another usage of the term “module” that I think may have some relevance to this whole discussion.   In evo-devo, modularity refers to the degree that a group of phenotypic characters have independent genetic architecture and ontogeny.  I will call this an “evolutionary ontogenetic module” (EOM) and contrast that with an “evolutionary cognitive module” (ECM) of EP.  Sperber (2002), in his defense of massive modularity, actually discusses EOMs in passing.  Pigliucci (2008) details the various, largely divergent definitions of modularity.  I tend to think about EOMs the way that Wagner & Altberg (1996) do, wherein a modular set of traits is one with (1) a higher than average level of integration by pleiotropic effects (i.e., gene interactions) and (2) a higher than average level of independence from other trait sets.  That is, modular architecture occurs where there are few pleiotropic genes that act across characters with different functions but more such effects falling on functionally related traits.

Modularity in the evo-devo sense is central to the evolution of complexity as well as the evolution of evolvability (the capacity of an organism to respond adaptively to selection).  Do ECMs need to be EOMs? Does this and other related concepts from evo-devo help provide a means for relating the ideas of EP or HBE to their genetic architecture and ontogenetic assembly?  I think so but I think an elaboration on this topic awaits a later post.

References

Barrett, H. C., and R. Kurzban. 2006. Modularity in Cognition: Framing the Debate. Psychological Review 113 (3):628-647.

Bird, R. B., and E. A. Smith. 2005. Signaling Theory, Strategic Interaction, and Symbolic Capital. Current Anthropology 46 (2):221-248.

Fodor, J. 1983. The Modularity of Mind. Cambridge: MIT Press.

Machery, E. 2007. Massive Modularity and Brain Evolution. Philosophy of Science74: 825–838.

Pigliucci, M. 2008. Is Evolvability Evolvable? Nature Genetics 9:75-82.

Pinker, S. 1997. How the Mind Works. New York: Norton.

Sperber, D. 2002. In Defense of massive modularity. In Dupoux, E.  Language, Brain and Cognitive Development: Essays in Honor of Jacques Mehler. Cambridge, Mass. MIT Press. 47-57.

Symons, D. 1989. A Critique of Darwinian Anthropology. Ethology and Sociobiology 10 (1-3):131-144.

Wagner, G.P., and L. Altenberg. 1996. Perspective: Complex Adaptations and the Evolution of Evolvability. Evolution 50 (3):967-976.