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19
Effect of neutral selection on the evolution of molecular species
 In Proc. R. Soc. London B
, 1998
"... We introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequencestructure maps are present also in ou ..."
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We introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequencestructure maps are present also in our general model. Examples are the occurrence of “common ” structures which occupy a fraction of the genotype space which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks which cover the genotype space in such a way that a member of such a network can be found within a small radius of any point in the space. We also describe a number of new phenomena which appear to be general properties of neutrally evolving systems. In particular, we show that the maximum fitness attained during the adaptive walk of a population evolving on such a fitness landscape increases with increasing degree of neutrality, and is directly related to the fitness of the most fit percolating network. 1
Modelling ’evodevo’ with RNA
 Bioessays
, 2002
"... Introduction Phenotype refers to the physica , organizationa and behaviora expression of an organismduani its ifetime. Genotype refers to a heritab e repository of information that instruYl the produlTj" of moecu es whose interactions, in conjujlzTE with the environment, generate and maintain t ..."
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Cited by 24 (0 self)
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Introduction Phenotype refers to the physica , organizationa and behaviora expression of an organismduani its ifetime. Genotype refers to a heritab e repository of information that instruYl the produlTj" of moecu es whose interactions, in conjujlzTE with the environment, generate and maintain the phenotype. The processes inking genotype to phenotype are known as deve opment. They intervene in the genesis of phenotypic nove ty from genetic mu tation. EvouGYYYxl trajectories therefore depend on deve opment. In tu(( evouG""""l processes shape deve opment, creating a feedback known as "evodevo" 1,2 . The main thruj of this review is to show that some key aspects of this feedback are present even in the microcosm of RNA fo ding. In a narrow sense, the re ation between RNA sequY"jl and their shapes is treated as a prob em in biophysics. Yet, in a wider sense, RNA fo ding can be regarded as a minima mode of a genotypephenotype re ation . The RNA mode is not a representation of organis
Asymptotic enumeration of RNA structures with pseudoknots
, 2007
"... In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for knoncrossing RNA structures. Our results are based on the generating function for the number of ..."
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Cited by 8 (5 self)
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In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for knoncrossing RNA structures. Our results are based on the generating function for the number of knoncrossing RNA pseudoknot structures, Sk(n), derived in [17], where k − 1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function P n≥0 Sk(n)z n and obtain for k = 2 and k = 3 the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2 and 3noncrossing RNA structures. Our main result is the derivation of the formula
A COMBINATORIAL FRAMEWORK FOR RNA TERTIARY INTERACTION
, 710
"... Abstract. In this paper we show how to express RNA tertiary interactions via the concept of tangled diagrams. Tangled diagrams allow to formulate RNA base triples and pseudoknotinteractions and to control the maximum number of mutually crossing arcs. In particular we study two subsets of tangled dia ..."
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Abstract. In this paper we show how to express RNA tertiary interactions via the concept of tangled diagrams. Tangled diagrams allow to formulate RNA base triples and pseudoknotinteractions and to control the maximum number of mutually crossing arcs. In particular we study two subsets of tangled diagrams: 3noncrossing partitions and braids. Our main results are asymptotic formulas for 3nocrossing partitions and braids derived by the analytic theory of singular difference equations due to BirkhoffTrjitzinsky. Explicitly, we prove for the number of 3noncrossing partitions and braids the formulas p3(n) ∼ K1 9 n n −7 (1 + t1/n + t2/n 2 + t3/n 3) and ρ3(n) ∼ K 8 n n −7 (1 + c1/n + c2/n 2 + c3/n 3), respectively where K, K1, ci, ti, i = 1, 2, 3 are constants. 1.
Sequence Redundancy in Biopolymers  A Study on RNA and Protein Structures
, 1997
"... Mapping sequences onto biopolymer structures is characterized by redundancy since the numbers of sequences exceed the numbers of structures. The degree of Redundancy depends on the notion of structure. Two classes of biopolymers, RNA molecules and proteins are considered in detail. A general feature ..."
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Cited by 5 (3 self)
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Mapping sequences onto biopolymer structures is characterized by redundancy since the numbers of sequences exceed the numbers of structures. The degree of Redundancy depends on the notion of structure. Two classes of biopolymers, RNA molecules and proteins are considered in detail. A general feature of sequence to structure mappings is the existence of a few common and many rare structures. Consequences of redundancy and frequency distribution of RNA structures are shape space covering and the existence of extended neutral networks. Populations migrate on neutral networks by a diffusionlike mechanism. Neutral networks are of fundamental importance for evolutionary optimization since they enable populations to escape from local optima of fitness landscapes.
C.: Inverse folding of RNA pseudoknot structures. Algorithms for Molecular Biology 5(27
, 2010
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Modular, knoncrossing diagrams
, 2010
"... In this paper we compute the generating function of modular, knoncrossing diagrams. A knoncrossing diagram is called modular if it does not contain any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA tertiary structures and their prope ..."
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In this paper we compute the generating function of modular, knoncrossing diagrams. A knoncrossing diagram is called modular if it does not contain any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA tertiary structures and their properties reflect basic features of these biomolecules. The particular case of modular noncrossing diagrams has been extensively studied. Let Qk(n) denote the number of modular knoncrossing diagrams over n vertices. We derive exact enumeration results as well for k = 3,...,9 and derive a new proof of the formula Q2(n) ∼ 1.4848n −3/2 1.8489n (Hofacker et al. 1998). as the asymptotic formula Qk(n) ∼ ckn−(k−1)2 − k−1 2 γ −n k 1
Combinatorics on Plane Trees, Motivated by RNA Secondary Structure Configurations
"... Motivated by the base pairing of RNA sequences, and as part of our collaboration with A. Condon and H. H. Hoos, we defined the local move operation on two unobstructed edges in a plane tree given here. We now consider the graph Gn induced by this operation on Tn, the set of plane trees with n edges. ..."
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Motivated by the base pairing of RNA sequences, and as part of our collaboration with A. Condon and H. H. Hoos, we defined the local move operation on two unobstructed edges in a plane tree given here. We now consider the graph Gn induced by this operation on Tn, the set of plane trees with n edges. We provide a series of results showing that Gn is a connected, npartite graph of diameter n − 1 with disjoint sets whose cardinalities are enumerated by the Narayana numbers. Our partition of Gn depends on orienting the edges in a plane tree, and differs from the known Narayana decomposition of ordered trees according to the number of leaves by N. Dershowitz and S. Zaks. We then consider the partial ordering � induced by an antisymmetric restriction of our local move operation. After a theorem characterizing the new � relation, we prove that the poset (Tn, �) is a lattice, which is complemented but not distributive. Finally, in a joint result with S. Fomin, we show that (Tn, �) is isomorphic to the lattice of noncrossing partitions. 1
CROSSINGS AND NESTINGS IN TANGLEDDIAGRAMS
, 710
"... Abstract. A tangleddiagram over [n] = {1,..., n} is a graph of degree less than two whose vertices 1,..., n are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen et.al. we prove a ..."
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Abstract. A tangleddiagram over [n] = {1,..., n} is a graph of degree less than two whose vertices 1,..., n are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen et.al. we prove a bijection between generalized vacillating tableaux with less than k rows and knoncrossing tangleddiagrams and study their crossings and nestings. We show that the number of knoncrossing and knonnesting tangleddiagrams are equal and enumerate tangleddiagrams. 1.