Results 1  10
of
97
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 additive const ..."
Abstract

Cited by 89 (15 self)
 Add to MetaCart
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.
Broken replica symmetry bounds in the mean field spin glass model
 Comm. Math Phys
, 2003
"... By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy w ..."
Abstract

Cited by 79 (11 self)
 Add to MetaCart
By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the SherringtonKirkpatrick model, and the Derrida pspin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric SherringtonKirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.
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 ..."
Abstract

Cited by 70 (29 self)
 Add to MetaCart
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...
The Thermodynamic Limit in Mean Field Spin Glass Models
 Commun. Math. Phys
, 2002
"... We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the SherringtonKirkpatrick model, and the Derrida pspin model. The main argument is based on a smoo ..."
Abstract

Cited by 68 (18 self)
 Add to MetaCart
We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the SherringtonKirkpatrick model, and the Derrida pspin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N1 and N2 sites, respectively, with N1 + N2 = N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site.
Landscapes And Molecular Evolution
, 1996
"... that allows to choose the direction for the next step at random from all directions along which fitness does not decrease. Stationary states of populations correspond to local optima of the fitness landscape. Evolution is seen as a series of transitions between optima with increasing fitness values. ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
that allows to choose the direction for the next step at random from all directions along which fitness does not decrease. Stationary states of populations correspond to local optima of the fitness landscape. Evolution is seen as a series of transitions between optima with increasing fitness values. Wright's metaphor saw a recent revival when sufficiently simple models of fitness landscapes became available [1, 41]. These models are based on spin glass theory [63, 66] or closely related to it like Kauffman's Nk model [42]. Evolution of RNA molecules has been studied by more realistic models that deal explicitly with molecular structures obtained from folding RNA sequences [23, 24]. Fitness values serving as input parameters for evolutionary dynamics were derived through evaluation of the structures. The complexity of RNA fitness landscapes originates from conflicting consequences of structural changes that are reminiscent of "frustration" in the theory of spin glasses [2]. Fitness in t
The Dynamics of a Genetic Algorithm for Simple Random Ising Systems
 Physica D
, 1996
"... A formalism is presented for analysing Genetic Algorithms. It is used to study a simple Genetic Algorithm consisting of selection, mutation and crossover which is searching for the ground states of simple random Isingspin systems: a randomfield ideal paramagnet and a spinglass chain. The formalis ..."
Abstract

Cited by 38 (17 self)
 Add to MetaCart
A formalism is presented for analysing Genetic Algorithms. It is used to study a simple Genetic Algorithm consisting of selection, mutation and crossover which is searching for the ground states of simple random Isingspin systems: a randomfield ideal paramagnet and a spinglass chain. The formalism can also be applied to other population based search techniques and to biological models of microevolution. To make the problem tractable, it is assumed that the population dynamics can be described by a few macroscopic order parameters and that the remaining microscopic degrees of freedom can be averaged out. The macroscopic quantities that are used are the cumulants of the distribution of fitnesses (or energies) in the population. A statistical mechanics model is presented which describes the population configuration in terms of the cumulants, this is used to derive equations of motion for the cumulants. Predictions of the theory are compared with experiments and are shown to predict th...
A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms
 Lecture Notes in Computer Science 865
, 1994
"... : A statistical mechanical formulation of the dyamics of genetic algorithms is described. This formulation allows the derivation of equations which predict the distributions of fitness with the population at one generation in terms of the distribution at the previous generation. The effects of selec ..."
Abstract

Cited by 33 (8 self)
 Add to MetaCart
: A statistical mechanical formulation of the dyamics of genetic algorithms is described. This formulation allows the derivation of equations which predict the distributions of fitness with the population at one generation in terms of the distribution at the previous generation. The effects of selection are problem independent, and the formulation predicts an optimal value of selection. Crossover and mutation are discussed in terms of a test problem  search for the low energy states of a random spin chain. The theory is compared with simulations and the agreement is good. 1 Introduction The effectiveness of genetic algorithms can depend crucially on how they are carried out. In order to apply a genetic algorithm effectively, two types of decisions must be made. The first is the choice of representation  solutions to the problem must be represented as strings and genetic search operators (e.g. crossover, selection) must be chosen appropriately to this. The second is the choice of p...
Replica Field Theory for Deterministic Models (II): A NonRandom Spin Glass with Glassy Behavior
, 1994
"... We introduce and study a model which admits a complex landscape without containing quenched disorder. Continuing our previous investigation we introduce a disordered model which allows us to reconstruct all the main features of the original phase diagram, including a low T spin glass phase and a com ..."
Abstract

Cited by 33 (11 self)
 Add to MetaCart
We introduce and study a model which admits a complex landscape without containing quenched disorder. Continuing our previous investigation we introduce a disordered model which allows us to reconstruct all the main features of the original phase diagram, including a low T spin glass phase and a complex dynamical behavior. condmat/9406074 ROM2F/94/016 RomaLa Sapienza 1027 1 Introduction In a recent companion paper [1] (which in the following we will quote as (A)) we have started (at the same time than Jean Philippe Bouchaud and Marc Mezard in [2]) a study of the role of replica field theory when applied to the study of systems which do not contain quenched disorder (for further connected work which helps clarifying this issue see [3, 4]). The immediate starting point which prompted our investigation (A) was a model of binary sequences with low autocorrelation, as originally discussed from Golay and Bernasconi [5, 6]. The model was for us a prototype of a system which does not cont...
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 statistical ..."
Abstract

Cited by 31 (16 self)
 Add to MetaCart
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...
Modelling Evolving Populations
 J. Theor. Biol
, 1996
"... A formalism is presented for modelling the evolutionary dynamics of a population of gene sequences. The formalism was originally developed for describing genetic algorithms. In this paper the formalism is elaborated by considering the evolution of an ensemble of populations. This allows the evolu ..."
Abstract

Cited by 28 (9 self)
 Add to MetaCart
A formalism is presented for modelling the evolutionary dynamics of a population of gene sequences. The formalism was originally developed for describing genetic algorithms. In this paper the formalism is elaborated by considering the evolution of an ensemble of populations. This allows the evolution to be modelled more accurately. To illustrate the formalism the problem of a population of gene sequences evolving in a multiplicative fitness landscape is considered. A comparison with simulations is made and shows very good agreement. More complicated problems have already been investigated including sexual recombination and evolution in a multivalleyed fitness landscape. These results will be briefly reviewed. 1 Introduction 1.1 The formalism In this paper we present a formalism for calculating the evolutionary dynamics of a population of gene sequences. The population evolves through a series of selection steps followed by modifications of the gene sequence. The fitness of ...