Holey adaptive landscapes

September 8, 2011 by Julian Xue

Hello world :-)

My research interests has veered off pure EGT, but my questions still center around evolution — particularly the evolution of complex systems that are made up of many small components working in unison. In particular, I’ve been studying Gavrilets et al. ‘s model of holey fitness landscapes, I think it’s a model with great potential for studying macroevolution, or evolution on very long timescales. I’m not the first one to this idea, of course — Arnold and many others have seen the possible connection also, although I think of it in a rather different light.

In this first post, I will give a short summary of this model, cobbled together from several papers and Gavrilets’ book, the Origin of Species. The basic premise is that organisms can be characterized by a large number of traits. When we say large, we mean very large — thousands or so. Gavrilets envisions this as being the number of genes in an organism, so tens of thousands. The important thing is that each of these traits can change independently of other ones.

The idea that organisms are points in very high dimensional space is not new, Fisher had it in his 1930 classic Genetical Theory of Natural Selection, where he used this insight to argue for micromutationism — in such high dimensional space, most mutations of appreciable size are detrimental, so Fisher argued that most mutations must be small (this result was later corrected by Kimura, Orr and others, who argued that most mutations must be of intermediate size, since tiny mutations are unlikely to fix in large populations).

However, even Fisher didn’t see another consequence of high-dimensional space, which Gavrilets exploited mercilessly. The consequence is that in high-enough dimensional space, there is no need to cross fitness valleys to move between one high fitness phenotype to another; all high fitness genotypes are connected. This is because connectivity is exceedingly easy in high dimensional space. Consider two dimensions, to get from one point to another, there are only two directions to move in. Every extra dimension offers a new option for such movement, that’s why there’s a minimum dimensionality to chaotic behavior — we can’t embed a strange attractor in a two dimensional phase plane, since trajectories can’t help but cross each other. Three dimensions is better, but n-dimensional space, where n is in the tens of thousands — that’s really powerful stuff.

Basically, every phenotype — every point in n-D space, is connected to a huge number of other points in n-D space. That is, every phenotype has a huge number of neighbors. Even if the probability of being a highly fit organism is exceedingly small, chances are high that one would exist among this huge number of neighbors. We know that if each highly fit phenotype is, on average, connected to another highly fit phenotype (via mutation), then the percolation threshold is reached where almost all highly fit phenotypes are connected in one giant connected component. In this way, evolution does not have to traverse fitness minima.

If we consider mutations to be point mutations of genes, then mutations can be considered to be a Manhattan distance type walk in n-D space. That’s just a fancy way of saying that we have n genes, and only one can be changed at a time. In that case, the number of neighbors any phenotype has is n, and if the probability of being highly fit is better than 1/n, then highly fit organisms are connected. This is even easier if we consider mutations to be random movements in n-D space. That is, if we consider an organism to be characterized by , where is the trait, and a mutation from results in ), such that is a random small number that can be negative, and the Euclidean distance between and is less than , where is the maximum mutation size, then the neighbors of fill up the volume of a ball of radius around . The volume of this ball grows exponentially with n, so even a tiny probability of being highly fit will find some neighbor of that is highly fit, because of the extremely large volume even for reasonably sized n.

The fact that evolution may never have to cross fitness minima is extremely important, it means that most of evolution may take place on “neutral bands”. Hartl and Taube had foreseen this really interesting result. Gavrilets mainly used this result to argue for speciation, which he envisions as a process that takes place naturally with reproductive isolation and has no need for natural selection.

Several improvements over the basic result have been achieved, mostly in the realm of showing that even if highly fit phenotypes are highly correlated (forming “highly fit islands” in phenotype space), the basic result of connectivity nevertheless holds (i.e. there will be bridges between those islands). Gavrilets’ book summarizes some early results, but a more recent paper (Gravner et al.) is a real tour-de-force in this direction. Their last result shows that the existence of “incompatibility sets”, that is, sets of traits that destroy viability, nevertheless does not punch enough holes in n-D space to disconnect it. Overall, the paper shows that even with correlation, percolation (connectedness of almost all highly fit phenotypes) is still the norm.

Next Thursday, I will detail some of my own criticisms to this model and its interpretation. The week after next, I will hijack this model for my own purposes and I will attempt to show that such a model can display a great deal of historical contingency, leading to irreversible, Muller’s Ratchet type evolution that carries on in particular directions even against fitness considerations. This type of model, I believe, will provide an interesting bridge between micro and macroevolution.