Holey Fitness Landscapes

The dimensionality of sequence space can be defined as the number of new sequences (DNAs, RNAs, or proteins) one can get from a sequence by changing single elements of the sequence. Even the simplest organisms known have on the order of thousand genes and on the order of million DNA base pairs. Each of the genes can be at at least several different states (known as alleles). Thus, the dimensionality of genotype space is at least on the order of thousands. It is on the order of millions if one considers DNA base pairs instead of genes. This results in an astronomically large number of possible genotypes (or DNA sequences) which is much higher than the number of organisms present at any given time or even cumulatively since the origin of life. There is an important consequence of this observation. Because of the redundancy in the genotype-fitness map, different genotypes are bound to have very similar (identical from any practical point of view) fitnesses. Unless there is a strongly 'nonrandom' assignment of fitnesses (say all well-fit genotypes are put together in a single 'corner' of the genotype space), a possibility exists that well-fit genotypes might form connected clusters (or networks) that might extend to some degree throughout the genotype space. If this were so, populations might evolve along these clusters by single substitutions and diverge genetically without going through any adaptive valleys.

The huge dimensionality of most biologically interesting fitness landscapes brings some new properties which one does not observe in low-dimensional landscapes (e.g., in 2D or 3D geographic landscapes). In particular, multidimensional landscapes are generically characterized by the existence of neutral and nearly neutral networks (also referred to as holey fitness landscapes) that extend throughout the landscapes and that can dramatically affect the evolutionary dynamics of the populations. A neutral network is a contiguous set of sequences possessing the same fitness. A nearly neutral network is a contiguous set of sequences possessing approximately the same fitness. A holey adaptive landscape is an adaptive landscape where relatively infrequent well-fit (or as Wright put it, 'harmonious') genotypes form a contiguous set that expands ('percolates') throughout the genotype space. An appropriate 3D image of such an adaptive landscape is a flat surface with many holes representing genotypes that do not belong to the percolating set (see Figure 1d).

As was discussed above, each of the three classical metaphors of fitness landscapes emphasizes certain features of the landscapes and evolutionary dynamics while neglecting or de-emphasizing others. Wright's metaphor of rugged fitness landscapes emphasizes the existence of multiple high-fitness combinations of genes and the need for stochastic factors to overcome selection for continuous evolution. The metaphor of a single-peak landscape reflects Fisher's belief that there is a single 'best' combination of genes and that selection alone is sufficient for evolutionary change and adaptation. The metaphor of flat fitness landscapes emphasizes Kimura's views on the importance of extensive genetic divergence by mutation and random genetic drift alone. In contrast, the metaphor of holey landscapes illustrated in Figure 1d emphasizes the percolating ridges of high-fitness genotypes at the expense of other features of multidimensional fitness landscapes.

Within the metaphor of holey landscapes, local adaptation and microevolution can be viewed as climbing from a hole towards a nearly neutral network ofgenotypes with fitnesses at a level determined by mutation-selection-

random drift balance. The process of climbing occurs on a shorter timescale than that necessary for speciation, clad diversification, and macroevolution. Once a ridge is reached, the population will be prevented by selection from slipping off this ridge to lower fitnesses and by mutation, recombination, and gene flow from climbing to higher fitnesses. Speciation occurs when a population evolves to a genetic state separated from its initial state by a hole. Holey fitness landscapes found numerous applications for studying speciation, innovations, evolvability, and robustness.

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