Local fitness landscape

During the screening, a large amount of fitness data is generated, but only the fitness information of the improved mutants is used to continue to the next round of evolution. The large ensemble of less fit mutants provides a view of the local fitness landscape. By analyzing these data, certain statistical landscape parameters can be... 

Fig. 11.12 Three sample fitness landscapes (a) has a single smooth maximum, (b) has many equivalent local maxima and one global maximum, but is circularly symmetric (c.) has many irregularly spaced local maxima, and is a good example of a rugged landscape.

Fig. 13. Example of a two-dimensional fitness landscape, x, and x2 are coordinates in a space. The global peak is the point of highest fitness in the space. A local peak is a point all of whose neighbors, defined by some neighbor relationship, are of lower fitness. In this figure, neighbors are points that differ by one in...

Random energy model The random energy model (REM) results from using a fitness distribution p(f) to assign fitnesses randomly to points in the landscape [ 14,59,60,70,71,81, 91,92], p(f) is the probability that a point in the sequence space has fitness fand is exactly analogous to affinity distribution p(Ka). Such landscapes have zero correlation (are very rugged), have many local fitness peaks, and result in very short adaptive walks. Very few of the local peaks are accessible by adaptive walks from any particular point. 

Fig. 4. The role of neutral networks in evolutionary optimization through adaptive walks and random drift. Adaptive walks allow to choose the next step arbitrarily from all directions where fitness is (locally) nondecreasing. Populations can bridge over narrow valleys with widths of a few point mutations. In the absence of selective neutrality (upper part) they are, however, unable to span larger Hamming distances and thus will approach only the next major fitness peak. Populations on rugged landscapes with extended neutral networks evolve along the network by a combination of adaptive walks and random drift at constant fitness (lower part). In this manner, populations bridge over large valleys and may eventually reach the global maximum ofthe fitness landscape.

Fig. 1. A two-dimensional projection of the hyperdimensional fitness landscape. In this simplified representation, sequence space is shown for a 4-mer where the colors represent amino acid types. The all-blue sequence is the global optimum the lower fitness peaks are local optima. The problem of in vitro evolution is how to search this space effectively, without becoming trapped at a suboptimal fitness.

Fig. 11. The local fitness distributions around fourteen representative wild types. The curves were determined analytically for the fully additive landscape by Aita and Husimi for sequence length N = 60 and alphabet size A = 20. Each wild type is shown at the center of the concentric circles. The axes y is the scaled fitness (= F/ sN, s is the mean of F and here is negative) and x is the scaled Hamming distance from the optimum (= do/N). Each local fitness distribution is expressed as a concentric pie chart showing the fraction of mutants having Ay between l/N and (/ + 1)/N, where l — — 5, —4, — 3,. . . , 4. The thick curves represent the contours satisfying Ay = 0. Reprinted from Aita and Husimi (1998a) with permission, 1998 by Academic Press.

Macken et al. (1991) identified two behaviors of an adaptive walk as it ascends the fitness landscape that are separated by a distinct phase transition. As a sequence becomes more optimized, the number of fitter mutants decreases. At some point, it becomes difficult to sample a fitter mutant given the number that can be screened. The probability of a sequence with fitness I having a single-mutant neighbor with lower fitness is G(F). The probability that a sequence with D single-mutant neighbors is at a local optimum 1 (F) is approximately... 

Ogata et al. attacked the same nucleic acid conformation problem, but replaced the buildup scheme of Lucasius with a local filter that is equivalent to the use of a rotamer library. In both cases, these methods must deal with the fact that this is an underconstrained problem because several of the dihedrals have no NOEs associated with them. Schuster earlier treated a simple model of RNA to predict three-dimensional (3D) conformations, using a variant on a spin-glass Hamiltonian as his fitness function. The simple model used allowed for the analysis of the complexity of the fitness landscape, couched in terms of the genotype-to-phenotype mapping. 

When approaching a new optimization problem one should first try to understand as much as possible about the nature of the fitness landscape. Is it smooth. Does it have only one or a few local minima Or alternatively is it very noisy, with many local minima If it is smooth, with only one or a small number of minima, then some method other than the GA (probably a local rather than a global method) is called for. If there is some rough way to characterize different regions of the fitness landscape, then it may be best to devise a fitness function using that information. This will increase the efficiency of the GA as well as that of any other method that might be used. 

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