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Funnels, Pathways and the Energy Landscape of Protein Folding: A Synthesis
 PROTEINS
, 1994
"... The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy lan ..."
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Cited by 154 (11 self)
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The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy landscape approach explains when and why unique behaviors, such as specific folding pathways, occur in some proteins and more generally explains the distinction between folding processes common to all sequences and those peculiar to individual sequences. This approach also gives new, quantitative insights into the interpretation of experiments and simulations of protein folding thermodynamics and kinetics. Specifically, the picture provides simple explanations for folding as a twostate firstorder phase transition, for the origin of metastable collapsed unfolded states and for the curved Arrhenius plots observed in both laboratory experiments and discrete lattice simulations. The relation of these quantitative ideas to folding pathways, to uniexponential vs. multiexponential behavior in protein folding experiments and to the effect of mutations on folding is also discussed. The success of energy landscape ideas in protein structure prediction is also described. The use of the energy landscape approach for analyzing data is illustrated with a quantitative analysis of some recent simulations, and a qualitative analysis of experiments on the folding of three proteins. The work unifies several previously proposed ideas concerning the mechanism protein folding and delimits the regions of validity of these ideas under different thermodynamic conditions.
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 ..."
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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...
On the Stability of the Quenched State in Mean Field Spin Glass Models
, 1997
"... While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying "quenched state". The assumption of such continuity in temperature implies that in the infinite volume limit ..."
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Cited by 82 (4 self)
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While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying "quenched state". The assumption of such continuity in temperature implies that in the infinite volume limit the state is stable under a class of deformations of the Gibbs measure. The condition is satisfied by the Parisi Ansatz, along with an even broader stationarity property. The stability conditions have equivalent expressions as marginal additivity of the quenched free energy. Implications of the continuity assumption include constraints on the overlap distribution, which are expressed as the vanishing of the expectation value for an infinite collection of multioverlap polynomials. The polynomials can be computed with the aid of a realreplica calculation in which the number of replicas is taken to zero. Key words: Mean field, spin glass, quenched state, overlap distribution, replicas. 1. Introducti...
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Multimeme Algorithms for Protein Structure
 In: Proceedings of the Parallel Problem Solving from Nature VII. Lecture Notes in Computer Science
, 2002
"... Despite intensive studies during the last 30 years researchers are yet far from the \holy grail" of blind structure prediction of the three dimensional native state of a protein from its sequence of amino acids. ..."
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Cited by 41 (16 self)
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Despite intensive studies during the last 30 years researchers are yet far from the \holy grail" of blind structure prediction of the three dimensional native state of a protein from its sequence of amino acids.
Characterization of invariant measures at the leading edge for competing particle systems Ann. Probab. 33 (2005), 82–113 (Francis Comets) Université Paris Diderot  Paris 7, Mathématiques, case 7012, F75 205 Paris Cedex 13, France Email address: comets@
"... We study systems of particles on a line which have a maximum, are locally finite, and evolve with independent increments. ‘Quasistationary states ’ are defined as probability measures, on the σ algebra generated by the gap variables, for which the joint distribution of the gaps is invariant under t ..."
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Cited by 39 (2 self)
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We study systems of particles on a line which have a maximum, are locally finite, and evolve with independent increments. ‘Quasistationary states ’ are defined as probability measures, on the σ algebra generated by the gap variables, for which the joint distribution of the gaps is invariant under the time evolution. Examples are provided by Poisson processes with densities of the form, ρ(dx) = e−sx s dx, with s> 0, and linear superpositions of such measures. We show that conversely: any quasistationary state for the independent dynamics, with an exponentially bounded integrated density of particles, corresponds to a superposition of the above described probability measures, restricted to the relevant σalgebra. Among the systems for which this question is of some relevance are spinglass models of statistical mechanics, where the point process represents the collection of the free energies of distinct “pure states”, the time evolution corresponds to the addition of a spin variable, and the Poisson measures described above correspond to the socalled REM states. 2 A. RUZMAIKINA and M. AIZENMAN 1.
The algebraic theory of recombination spaces
, 2000
"... A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "g ..."
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Cited by 35 (16 self)
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A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourierdecomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for Pstructures. For binary string recombination the elementary landscapes are exactly the pspin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...
Boolean Dynamics with Random Couplings
, 2002
"... This paper reviews a class of generic dissipative dynamical systems called NK models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends ..."
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Cited by 34 (0 self)
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This paper reviews a class of generic dissipative dynamical systems called NK models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.
Nonequilibrium fluctuations in small systems: From physics to biology
 Advances in Chemical Physics
, 2006
"... In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule fo ..."
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Cited by 19 (1 self)
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In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule force experiments. Next I present a general discussion on path thermodynamics and consider distributions of work/heat fluctuations as large deviation functions. Then I address the topic of glassy dynamics from the perspective of nonequilibrium fluctuations due to small cooperatively rearranging regions. Finally, I