NK Landscapes Research Articles

Epistasis in genetic algorithms revisited

Davidor epistasis is first analyzed to give its meaning a better understanding. Then, a generalization is introduced to investigate interactions between one or more positions on a chromosome, allowing a more accurate analysis of fitness landscape. A new probabilistic tool based on epistasis is also defined to compute a normalized measure of problem complexity. This tool, a graded correlation, is expressed in the Walsh basis to determine the fitness functions of minimal complexity with this measure. These results are applied to an example given by Manela and Campbell of a fully epistatic function that hides a linear structure. This function is proved to be of minimal complexity with respect to graded correlation. Then, a test is also performed on some classical fitness functions such as NK landscapes, and a problem of coding choice is finally solved.

Adaptive walks with noisy fitness measurements.

Adaptive walks constitute an optimization technique for searching a space of possible solutions, for example, a space of different molecules. The goal is to find a point in space (a molecule) that is optimal or near-optimal in some property, generally referred to as the 'fitness', such as its ability to bind to a given receptor. Adaptive walking, an analog of natural selection, is a powerful technique for searching landscapes. However, errors in the measurements will cause errors in the adaptive walks. Mutant molecules of higher fitness may be ignored or mutants of lower fitness may be accepted. To examine the effect of measurement error on adaptive walks, we simulate single-agent hill-climbing walks on NK landscapes of varying ruggedness where Gaussian noise is added to the fitness values to model measurement error. We consider both constant measurement noise and noise whose variance decays exponentially with fitness. We show that fitness-independent noise can cause walks to 'melt' off the peaks in a landscape, wandering in larger regions as the noise increases. However, we also show that a small amount of noise actually helps the walk perform better than with no noise. For walks in which noise decreases exponentially with fitness, the most characteristic behavior is that the walk meanders throughout the landscape until it stumbles across a point of relatively high fitness, then it climbs the landscape towards the nearest peak. Finally, we characterize the balance between selection pressure and noise and show that there are several classes of walk dynamic behavior.

Adaptation on Fluctuating Fitness Landscapes: Speciation and the Persistence of Lineages

The NK landscape model of adaptive evolution introduced by Kauffman & Levin (1987) is modified to simulate diversification of monophyletic lineages. Extinction probability is made a function of fitness, and speciation probability a function of the number of fitter one mutant neighbors, which is analogous to genetic potential. Dynamics of adaptive walks are measured as the total number of phenotypically defined species per generation. Changes in species number are described for static and varying landscapes. Persistence of lineages is incompatible with static fitness landscapes and requires that most component species do not reach adaptive optima. Differences in N :K ratios, and therefore in morphological integration, strongly affect speciation rates and could play a role in species sorting.

The NK model of rugged fitness landscapes and its application to maturation of the immune response

Adaptive evolution is, to a large extent, a complex combinatorial optimization process. Such processes can be characterized as "uphill walks on rugged fitness landscapes". Concrete examples of fitness landscapes include the distribution of any specific functional property such as the capacity to catalyze a specific reaction, or bind a specific ligand, in "protein space". In particular, the property might be the affinity of all possible antibody molecules for a specific antigenic determinant. That affinity landscape presumably plays a critical role in maturation of the immune response. In this process, hypermutation and clonal selection act to select antibody V region mutant variants with successively higher affinity for the immunizing antigen. The actual statistical structure of affinity landscapes, although knowable, is currently unknown. Here, we analyze a class of mathematical models we call NK models. We show that these models capture significant features of the maturation of the immune response, which is currently thought to share features with general protein evolution. The NK models have the important property that, as the parameter K increases, the "ruggedness" of the NK landscape varies from a single peaked "Fujiyama" landscape to a multi-peaked "badlands" landscape. Walks to local optima on such landscapes become shorter as K increases. This fact allows us to choose a value of K that corresponds to the experimentally observed number of mutational "steps", 6-8, taken as an antibody sequence matures. If the mature antibody is taken to correspond to a local optimum in the model, tuning the model requires that K be about 40, implying that the functional contribution of each amino acid in the V region is affected by about 40 others. Given this value of K, the model then predicts several features of "antibody space" that are in qualitative agreement with experiment: (1) The fraction of fitter variants of an initial "roughed in" germ line antibody amplified by clonal selection is about 1-2%. (2) Mutations at some sites of the mature antibody hardly affect antibody function at all, but mutations at other sites dramatically decrease function. (3) The same "roughed in" antibody sequence can "walk" to many mature antibody sequences. (4) Many adaptive walks can end on the same local optimum. (5) Comparison of different mature sequences derived from the same initial V region shows evolutionary hot spots and parallel mutations. All these predictions are open to detailed testing by obtaining monoclonal antibodies early in the immune response and carrying out in vitro mutagenesis and adaptive hill climbing with respect to affinity for the immunizing antigen.