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Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an addi ..."
Abstract

Cited by 105 (16 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Fitness Landscapes
 Appl. Math. & Comput
, 2002
"... . Fitness landscapes are a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space that is equipped with some notion of adjacency, nearness, distance or accessibility, into the real numbe ..."
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Cited by 83 (14 self)
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. Fitness landscapes are a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space that is equipped with some notion of adjacency, nearness, distance or accessibility, into the real numbers. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. This chapter gives a brief overview on recent developments in this area, focusing on "geometrical" properties of landscapes. 1 Introduction The concept of a fitness landscape originated in theoretical biology more than seventy years ago [1]. It can be thought of as a kind of "potential function" underlying the dynamics of evolutionary optimization. Implicit in this idea is both a fitness function f that assigns a fitness value to every possible genotype (or organism), and the arrangement of t...
RNA Folding and Combinatory Landscapes
, 1993
"... In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure. Considering only the free energy leads to an energy landscape over the sequence space. Taking into account structur ..."
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Cited by 76 (29 self)
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In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure. Considering only the free energy leads to an energy landscape over the sequence space. Taking into account structure generates a less visualizable nonscalar "landscape", where a sequence space is mapped into a space of discrete "shapes". We investigate the statistical features of both types of landscapes by computing autocorrelation functions, as well as distributions of energy and structure distances, as a function of distance in sequence space. RNA folding is characterized by very short structure correlation lengths compared to the diameter of the sequence space. The correlation lengths depend strongly on the size and the pairing rules of the underlying nucleotide alphabet. Our data suggest that almost every minimum free energy structure is found within a small neighborhood of any random sequence. The...
Fitness landscapes and evolvability
 Evolutionary Computation
, 2001
"... In this paper, we develop techniques based on evolvability statistics of the tness landscape surrounding sampled solutions. Averaging the measures over a sample of equal tness solutions allows us to build up tness evolvability portraits of the tness landscape, which we show can be used to compare ..."
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Cited by 52 (2 self)
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In this paper, we develop techniques based on evolvability statistics of the tness landscape surrounding sampled solutions. Averaging the measures over a sample of equal tness solutions allows us to build up tness evolvability portraits of the tness landscape, which we show can be used to compare both the ruggedness and neutrality in a set of tunably rugged and tunably neutral landscapes. We further show that the techniques can be used with solution samples collected through both random sampling of the landscapes and online sampling during optimization. Finally, we apply the techniques to two real evolutionary electronics search spaces and highlight differences between the two search spaces, comparing with the time taken to nd good solutions through search.
Memetic Algorithms for the Traveling Salesman Problem
 Complex Systems
, 1997
"... this paper, the tness landscapes of several instances of the traveling salesman problem (TSP) are investigated to illustrate why MAs are wellsuited for nding nearoptimum tours for the TSP. It is shown that recombination{based MAs can exploit the correlation structure of the landscape. A comparis ..."
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Cited by 37 (8 self)
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this paper, the tness landscapes of several instances of the traveling salesman problem (TSP) are investigated to illustrate why MAs are wellsuited for nding nearoptimum tours for the TSP. It is shown that recombination{based MAs can exploit the correlation structure of the landscape. A comparison of several recombination operators { including a new generic recombination operator { reveals that when using the sophisticated Lin{Kernighan local search, the performance dierence of the MAs is small. However, the most important property of eective recombination operators is shown to be respectfulness. In experiments it is shown that our MAs with generic recombination are among the best evolutionary algorithms for the TSP. In particular, optimum solutions could be found up to a problem size of 3795, and for large instances up to 85,900 cities, nearoptimum solutions could be found in a reasonable amount of time
Landscapes  Complex Optimization Problems and Biopolymer Structures
 Computers Chem
, 1993
"... The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by s ..."
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Cited by 33 (16 self)
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The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by statistical measures like number of optima, lengths of walks, and correlation functions. The evolution of a quasispecies on such landscapes exhibits three dynamical regimes depending on the replication fidelity: Above the "localization threshold" the population is centered around a (local) optimum. Between localization and "dispersion threshold" the population is still centered around a consensus sequence, which, however, changes in time. For very large mutation rates the population spreads in sequence space like a gas. The critical mutation rates separating the three domains depend strongly on characteristics properties of the fitness landscapes. Statistical characteristics of RNA landscapes are acces...
Statistics of RNA Melting Kinetics
, 1993
"... We present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns structure dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landsc ..."
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Cited by 31 (13 self)
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We present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns structure dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landscape" of reaction rates, or activation energies, over the space of sequences with fixed length. We study the distribution and the correlation structure of these activation energies. 1. Introduction Single stranded RNA sequences fold into complex threedimensional structures. A tractable, yet reasonable, model for the map from sequences to structures considers a more coarse grained level of resolution known as the secondary structure. The secondary structure is a list of base pairs such that no pairings occur between bases located in different loop regions. Algorithms based on empirical energy data have been developed to compute the minimum free energy secondary structure of an RNA sequence (Zuker...
Fitness landscapes and difficulty in genetic programming
 In Proceedings of the First IEEE Conference on Evolutionary Computing
, 1994
"... ..."
Why Some Fitness Landscapes are Fractal
, 1993
"... Many biological and biochemical measurements, e.g. the "fitness" of a particular genome, or the binding affinity to a particular substrate, can be treated as a "fitness landscape", an assignment of numerical values to points in sequence space (or some other configuration space). ..."
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Cited by 25 (11 self)
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Many biological and biochemical measurements, e.g. the "fitness" of a particular genome, or the binding affinity to a particular substrate, can be treated as a "fitness landscape", an assignment of numerical values to points in sequence space (or some other configuration space). As an alternative to the enormous amount of data required to completely describe such a landscape, we propose a statistical characterization, based on the properties of a random walk through the landscape, and, more specifically, its autocorrelation function. Under assumptions roughly satisfied by two classes of simple model landscapes (the Nk model and the pspin model) and by the landscape of estimated free energies of RNA secondary structures, this autocorrelation function, along with the mean and variance of individual points and the size of the landscape, completely characterize it. Having noted that these and other landscapes of estimated replication and degradation rates all have a well defined correlat...
Random Field Models For Fitness Landscapes
, 1996
"... In many cases fitness landscapes are obtained as particular instances of random fields by assigning a large number of random parameters. Models of this type are often characterized reasonably well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of thei ..."
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Cited by 21 (6 self)
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In many cases fitness landscapes are obtained as particular instances of random fields by assigning a large number of random parameters. Models of this type are often characterized reasonably well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of their Fourier series expansions and investigate the relation between the covariance matrix of the random field model and the correlation structure of the individual landscapes constructed from this random field. Our formalism suggests to approximate landscape with known autocorrelation function by a random field model that has the same correlation structure.