Reidys, C., Stadler, P., & Schuster, P. (1997). Generic properties of combinatory maps: neutral networks of rna secondary structures. Bulletin of Mathematical Biology, 59(2), 339--397.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... algorithm for this problem (Theorem 16) P11 Given a 3D structure, determine whether the fittest protein sequences are connected, i.e. whether they can mutate into each other through allowable mutations, such as point mutations, while the intermediate protein sequences all remain the fittest [2, 8, 17, 20, 22, 29, 31]. This problem takes O(#) time after a certain maximum network flow is computed (Theorem 11) P12 Given a 3D structure, in the case that the set of all fittest protein sequences is not connected, determine whether two given fittest protein sequences are connected. This problem takes O(#) time ....

C. Reidys, P. Stadler, and P. Schuster. Generic properties of combinatory maps: Neutral networks of RNA secondary structures. Bulletin of Mathematical Biology, 59:339--397, 1997.

.... Kimura on selectively neutral mutation initiated a debate amongst biologists which continues to this day (Kimura, 1983; Crow and Kimura, 1970) More recently, research into the structure of RNA secondary structure folding landscapes (Fontana et al. 1993; Forst et al. 1995; Huynen et al. 1996; Reidys et al. 1997) led to the concept of neutral networks. These are connected networks of genotypes which map to the same phenotype, where two genotypes are connected if they differ by one (or possibly a few) point mutations. It is found that the dynamics of populations of genotypes evolving on fitness ....

Reidys, C., Stadler, P. F., and Schuster, P. (1997). Generic properties of combinatory maps: Neutral networks of RNA secondary structures. Bull. Math. Biol., 59(2):339--397.

.... This work includes Kimura s neutral theory of molecular evolution (Kimura 1983) Eigen s analysis of the molecular quasispecies (Eigen, McCaskill and Schuster 1989; Nowak and Schuster 1989) and recent developments in the understanding of RNA evolution both in vitro, in simulation and analytically (Reidys, Stadler and Schuster 1997; Schuster et al. 1994; Baskaran, Stadler and Schuster 1996; Gr ner et al. 1996) Neutrality has also been detected in various protein models. In molecular biology it is clear that there is often a high degree of redundancy in the coding from genotype to phenotype there may indeed be ....

Reidys, C.M., Stadler, P.F. & Schuster, P. 1997. Generic Properties of Combinatory Maps - Neutral Networks of RNA Secondary Structures. Bull. Math. Biol. 59: 339397.

....dynamics. This awareness has thus far not extended to the GA community. Of particular interest is the notion of neutral networks of selectively neutral genotypes which percolate a fitness landscape recent work on RNA folding landscapes characterises their structure in terms of such networks [21, 6, 20, 12, 13, 1]. There is at present a lack of computationally tractable abstract models demonstrating neutrality. In this paper we introduce two parametrised families of abstract landscapes: the NKp landscapes, based on the NK family of abstract landscapes [15] allow tuning of the degree of neutrality whilst ....

....significant factor in evolutionary dynamics. This work includes Motoo Kimura s Neutral Theory of molecular evolution [16, 3] Manfred Eigen s analysis of molecular quasispecies [5, 2, 18] and recent developments in the understanding of RNA evolution both in vitro, in simulation and analytically [21, 22, 6, 20, 12, 13, 1]. A picture emerges of populations engaged not in hill climbing but rather drifting along connected networks of neutral genotypes, with sporadic jumps between networks. These neutral networks are of particular significance if they percolate the landscape i.e. they come arbitrarily close to ....

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Reidys, C.M., Stadler, P.F. & Schuster, P. (1995). Generic Properties of Combinatory Maps - Neutral Networks of RNA Secondary Structures. Santa Fe Institute Preprint 95-07-058, Santa Fe, NM, USA

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C. Reidys, P.F. Stadler, P. Schuster, "Generic properties of combinatory maps. Neutral networks of RNA secondary structures", Bull.Math.Biol., Vol.59, pp.339-397, 1997.

.... forming structure s k , G[s k ] f (s k ) fx j jf(x j ) s k g : 2) The neutral network G[s k ] is a graph on this set with edges connecting all pairs of sequences with Hamming distance one (figure1) For many purposes sequence structure maps can be approximated well by random graphs [45] which encapsulate generic features: All sequences folding into the same structure form a neutral network which is represented by a (random) graph. The random graph approach is based on a single parameter, which expresses the fraction of neutral neighbors averaged over all members of the ....

....pre image of sk in sequence space. Connecting all pairs of sequences with Hamming distance one yields a graph called the neutral network G[sk ] Neutral networks may be connected (A) or disconnected (B) Generic disconnected networks consist of a largest gaint component and many small islands [28, 29, 45]. the basis for applied molecular evolution, an already established new area of lively interest in biotechnology (See, for example, the special issue [64] the corresponding chapters in [21] and the recent review [66] In addition, the quasispecies model has been applied successfully also to ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....here. In the next section we shall introduce a particularly simple example of genotype phenotype mapping that is based on RNA secondary structures. In this case the mapping can be studied by computer simulation [40, 41] and analyzed mathematically by means of a model based on random graph theory [74]. 3. Genotype phenotype mapping with RNA secondary structures Already Sol Spiegelman [85] had pointed out that genotype and phenotype are two features of the same molecule in case of RNA evolution in the test tube, the The computation of the mutation matrix from the uniform error rate model is ....

....model system for the study of global relations between genotypes and phenotypes. The conventional onesequence one structure approach of structural biology is extended to a general concept that considers sequence structure relations as (non invertible) mappings from sequence space into shape space [34, 74, 82]. Application of combinatorics allows to derive an asymptotic expression for the numbers of acceptable structures as a function of the chain length n [47, 81] S n 1:4848 n 3=2 (1:8488) 4) This expression is based on two assumptions: i) the minimum stack length is two base pairs (n ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339-397, 1997.

....be a simple function of the distance between the phenotype and some target to be approached. Now we are in a position to classify different mappings: i) maps a discrete vector space, the sequence space I, into another nonscalar discrete (or continuous) space S. We call it a combinatory map [81] since the sequence space I is derived by a combinatory building principle (see also Sect. 3) ii) f( maps a discrete (or continuous) non scalar space S into the nonnegative real numbers IR . It represents an example of a landscape or cost function. In particular, f( is the fitness ....

....flexible phenotypes can respond to the appearance of new molecules in their environments. Explicit 20 Evolution of Phenotypes TABLE 1 Various strategies applied to study sequence structure maps of RNA Method Advantage Disadvantage Ref. Mathematical Random graph Analytical Limited validity of [81] model theory expressions model assumptions Exhaustive Folding algorithm Exact results Limited to short [39, 40] folding and and handling of chains: enumeration large samples up GC, 30 to 10 10 objects AUGC, 16 Statistical Inverse folding or Applicability Limited accuracy [29, 89] ....

[Article contains additional citation context not shown here]

Reidys, C., P. F. Stadler, and P. Schuster, "Generic Properties of Combinatory Maps. Neutral Networks of RNA Secondary Structure." Bull. Math. Biol. 59 (1997):339--397.

....provide convincing evidence that RNA landscapes are as simple as they could possibly be for evolutionary adaptation. The consequences for evolutionary optimization, the early stages of life, and molecular biotechnology are immediate. Based on these findings a random graph theory was developed [11] that explains the structure of neutral networks in terms of a single parameter: the frequency of neutral mutations. The predictions of this theory, among them connectedness and density of the neutral networks cannot be verified by a statistical approach based on sampling tiny fractions of ....

....Maps ffl How is f Gamma1 (s) embedded in sequence space Many of these questions have been at least partially answered in previous papers describing non exhaustive computer simulations [3, 9, 2, 5, 10, 4] as mentioned in the introduction. Based on these results a random graph model [11] was conceived that allows to construct the preimages of a (given) secondary structures s as functions of a single parameter, the (average) fraction of neutral neighbors of the sequences folding into s (see the section on neutral networks below) The random graph model reveals the generic ....

[Article contains additional citation context not shown here]

Christian Reidys, Peter Schuster, and Peter F. Stadler. Generic properties of combinatory maps --- neutral networks of rna secondary structures. Bull. Math. Biol., 1995. submitted; SFI-preprint 95-07-058.

....be a simple function of the distance between the phenotype and some target to be approached. Now we are in a position to classify different mappings: i) maps a discrete vector space, the sequence space I, into another nonscalar discrete (or continuous) space S. We call it a combinatory map [73] since the sequence space I is derived by a combinatory building principle (see also Sect. 3) ii) f( maps a discrete (or continuous) non scalar space S into the nonnegative real numbers IR . It represents an example of a landscape, in particular, it is the fitness landscape assigning a ....

....although the higher computational efforts may be critical for the current possibilities. 20 Evolution of Phenotypes TABLE 1 Various strategies applied to study sequence structure maps of RNA Method Advantage Disadvantage Ref. Mathematical Random graph Analytical Limited validity of [73] model theory expressions model assumptions Exhaustive Folding algorithm Exact results Limited to short [37, 38] folding and and handling of chains: enumeration large samples GC, 30 ( 10 9 objects) AUGC, 16 Statistical Inverse folding or Applicability Limited accuracy [27, 79] ....

[Article contains additional citation context not shown here]

Reidys, C., P. F. Stadler, and P. Schuster, "Generic Properties of Combinatory Maps. Neutral Networks of RNA Secondary Structure." Bull. Math. Biol. 59 (1997):339--397.

....structure. biology which is essentially concerned with the folding of a single sequence into one structure under minimum free energy conditions has to be extended to a global concept that considers sequence structure relations as (non invertible) mappings from sequence space into shape space [22, 52, 60]. Application of combinatorics allows to derive an asymptotic expression for the numbers of acceptable structures as a function of the chain length n [31, 58] S n 1:4848 Theta n 3=2 (1:8488) n : 7) This expression is based on two assumptions: i) the minimum stack length is two base pairs ....

....we ought to know also how sequences folding into the same structure are organized in sequence space. Sets of sequences forming the same structure have been called neutral networks. So far two approaches were applied to study neutral networks: a mathematical model based on random graph theory [52] and exhaustive folding [26] The mathematical model assumes that sequences forming the same structure are distributed randomly in the space of compatible sequences. A sequence is compatible with a structure when it can, in principle, fold into the structure in question. It has complementary bases ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 1995. Submitted. SFI-Preprint Series No. 95-07-058. 26 Peter Schuster/Landscapes and Molecular Evolution

.... structure s k , G[s k ] f Gamma1 (s k ) fx j jf(x j ) s k g : 2) The neutral network G[s k ] is a graph on this set with edges connecting all pairs of sequences with Hamming distance one (figure1) For many purposes sequence structure maps can be approximated well by random graphs [45] which encapsulate generic features: All sequences folding into the same structure form a neutral network which is represented by a (random) graph. The random graph approach is based on a single parameter, which expresses the fraction of neutral neighbors averaged over all members of the ....

....pre image of sk in sequence space. Connecting all pairs of sequences with Hamming distance one yields a graph called the neutral network G[sk ] Neutral networks may be connected (A) or disconnected (B) Generic disconnected networks consist of a largest gaint component and many small islands [28, 29, 45]. Reidys, Forst, Schuster: Replication on Neutral Networks 8 the basis for applied molecular evolution, an already established new area of lively interest in biotechnology (See, for example, the special issue [64] the corresponding chapters in [21] and the recent review [66] In addition, the ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....here. In the next section we shall introduce a particularly simple example of genotype phenotype mapping that is based on RNA secondary structures. In this case the mapping can be studied by computer simulation [40, 41] and analyzed mathematically by means of a model based on random graph theory [74]. 3. Genotype phenotype mapping with RNA secondary structures Already Sol Spiegelman [85] had pointed out that genotype and phenotype are two features of the same molecule in case of RNA evolution in the test tube, the 2 The computation of the mutation matrix from the uniform error rate model ....

....model system for the study of global relations between genotypes and phenotypes. The conventional onesequence one structure approach of structural biology is extended to a general concept that considers sequence structure relations as (non invertible) mappings from sequence space into shape space [34, 74, 82]. Application of combinatorics allows to derive an asymptotic expression for the numbers of acceptable structures as a function of the chain length n [47, 81] Sn 1:4848 Theta n 3=2 (1:8488) n : 4) This expression is based on two assumptions: i) the minimum stack length is two base pairs ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

.... is tantamount to the mapping of RNA sequences onto molecular structures which are approximated here by coarse grained and simplified versions, the so called secondary structures (represented by listings of Watson Crick and GU base pairings) Following the ideas of Christian Reidys et al. [32] sequence structure maps can be approximated very well by random graphs. All sequences folding into the same structure form a neutral network which is represented by a (random) graph. These networks represent the essential objects in the RNA folding landscape since they constituate the elements of ....

....RNA sequences which increase with 4 n or 2 n , respectively. The mapping from sequences into (secondary) is there fore highly redundant or many to one. Random graph theory applied to hypercubes [33] provides useful tools for an analytical approach to these redundant mappings of RNA molecules [32]. Here we present shortly the results which are important for the forthcoming analysis. 2.1. RNA secondary structures and compatible sequences Throughout this contribution we shall assume that the chain length of RNA molecules remains unchanged. Then the space of all sequences is the generalized ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....notion of structure which can be predicted from sequences by fast algorithms. Our work is, therefore, intermediate in abstraction: it approximates an empirical situation, while aiming at generally valid regularities which may serve as axioms for more abstract mathematical models (see, for example, [15]) 2. Generic properties of folding The term folding is used here to denote a surjection f : S 7 Sigma from the set S of all sequences of fixed length over the AUGC alphabet onto the set Sigma of all minimum free energy (mfe) secondary structures for that length. The map is established ....

.... in the set of empirical energy parameters or the thermodynamic level of description (one mfe structure versus the Boltzmann ensemble for a given sequence) 2 [20] These statistical properties, in particular neutrality, have led to a mathematical model based on percolation in random graphs [15]. In recent work we began to link the properties of the folding map with features observed in evolutionary dynamics. Model RNA populations in a flow reactor were subject to selection for a prespecified target structure. In particular, we found diffusion of the population on a neutral network, and ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....notion of structure which can be predicted from sequences by fast algorithms. Our work is, therefore, intermediate in abstraction: it approximates an empirical situation, while aiming at generally valid regularities which may serve as axioms for more abstract mathematical models (see, for example, [15]) 2. Generic properties of folding The term folding is used here to denote a surjection f : S ## # from the set S of all sequences of fixed length over the AUGC alphabet onto the set # of all minimum free energy (mfe) secondary structures for that length. The map is established implicitly ....

.... in the set of empirical energy parameters or the thermodynamic level of description (one mfe structure versus the Boltzmann ensemble for a given sequence) 2 [20] These statistical properties, in particular neutrality, have led to a mathematical model based on percolation in random graphs [15]. In recent work we began to link the properties of the folding map with features observed in evolutionary dynamics. Model RNA populations in a flow reactor were subject to selection for a prespecified target structure. In particular, we found di#usion of the population on a neutral network, and ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....notion of structure which can be predicted from sequences by fast algorithms. Our work is, therefore, intermediate in abstraction: it approximates an empirical situation, while aiming at generally valid regularities which may serve as axioms for more abstract mathematical models (see, for example, [17]) 2. Generic properties of folding The term folding is used here to denote a surjection f : S ## # from the set S of all sequences of fixed length over the AUGC alphabet onto the set # of all minimum free energy (mfe) secondary structures for that length. The map is established implicitly ....

.... in the set of empirical energy parameters or the thermodynamic level of description (one mfe structure versus the Boltzmann ensemble for a given sequence) 2 [22] These statistical properties, in particular neutrality, have led to a mathematical model based on percolation in random graphs [17]. In recent work we began to link the properties of the folding map with features observed in evolutionary dynamics. Model RNA populations in a flow reactor were subject to selection for a prespecified target structure. In particular, we found di#usion of the population on a neutral network, and ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997.

....provide convincing evidence that RNA landscapes are as simple as they could possibly be for evolutionary adaptation. The consequences for evolutionary optimization, the early stages of life, and molecular biotechnology are immediate. Based on these findings a random graph theory was developed [11] that explains the structure of neutral networks in terms of a single parameter: the frequency of neutral mutations. The predictions of this theory, among them connectedness and density of the neutral networks cannot be verified by a statistical approach based on sampling tiny fractions of ....

....open structure ffl How is f Gamma1 (s) embedded in sequence space Many of these questions have been at least partially answered in previous papers describing non exhaustive computer simulations [3, 9, 2, 5, 10, 4] as mentioned in the introduction. Based on these results a random graph model [11] was conceived that allows to construct the preimages of a (given) secondary structures s as functions of a single parameter, the (average) fraction of neutral neighbors of the sequences folding into s (see the section on neutral networks below) The random graph model reveals the generic ....

[Article contains additional citation context not shown here]

C. Reidys, P.F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 1995. Submitted. SFI-Preprint Series No. 95-07-058.

....the study of global relations between genotypes and phenotypes. The conventional approach of structural biology determining structures for single sequences is extended to a general concept that considers sequence structure relations as (non invertible) mappings from sequence space into shape space [17, 55, 60]. 2.2. Common and rare RNA structures Application of combinatorics to RNA secondary structures [72] allows to derive an asymptotic expression from a simple recursion for the numbers of (acceptable) structures that can be formed by sequences of chain length n [32, 59] S n 1:4848 Theta n 3=2 ....

....are neutral in the sense that the corresponding sequences fold into the same secondary structure. Detailed data can be found in [26] Two approaches have been applied so far to study the topology of neutral sets: a mathematical model of genotype phenotype mapping based on random graph theory [55] and exhaustive folding of all sequences with given chain length n [27] The mathematical model assumes that sequences forming the same structure are distributed randomly (in the space of compatible sequence, see below) and it uses the fraction of neutral neighbors as (the only) input parameter. ....

[Article contains additional citation context not shown here]

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps --- Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339--397, 1997. -- 28 -- Schuster & Stadler: Redundancy in Biopolymers

....notion of structure which can be predicted from sequences by fast algorithms. Our work is, therefore, intermediate in abstraction: it approximates an empirical situation, while aiming at generally valid regularities which may serve as axioms for more abstract mathematical models (see, for example, [15]) 2. Generic properties of folding The term folding is used here to denote a surjection f : S###fromtheset Sof all sequences of fixed length over the AUGC alphabet onto the set # of all minimum free energy (mfe) secondary structures for that length. The map is established implicitly by a ....

.... in the set of empirical energy parameters or the thermodynamic level of description (one mfe structure versus the Boltzmann ensemble for a given sequence) 2 [20] These statistical properties, in particular neutrality, have led to a mathematical model based on percolation in random graphs [15]. In recent work we began to link the properties of the folding map with features observed in evolutionary dynamics. Model RNA populations in a flow reactor were subject to selection for a prespecified target structure. In particular, we found di#usion of the population on a neutral network, and ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bul l. Math. Biol., 59:339--397, 1997.

....j f Gamma1 j = 88811 in 804 components 4723 . 0:286 0:373 j f Gamma1 j = 58580 in 649 components 13135 ( 0:214 0:246 j f Gamma1 j = 13737 in 503 components A graph of the distribution of component sizes can be found in [4]. y Very small components are not shown in detail here. A number in square brackets gives the total number of sequences in them. Data are taken from reference [5] 10 Gr uner et al. Analysis of RNA Sequence Structure Maps 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 lu 0.0 0.1 0.2 0.3 ....

C. Reidys, P.F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 1995. Submitted. SFI-Preprint Series No. 95-07-058.

....notion of structure which can be predicted from sequences by fast algorithms. Our work is, therefore, intermediate in abstraction: it approximates an empirical situation, while aiming at generally valid regularities which may serve as axioms for more abstract mathematical models (see, for example, [17]) 2. Generic properties of folding The term folding is used here to denote a surjection f : S 7 from the set S of all sequences of xed length over the AUGC alphabet onto the set of all minimum free energy (mfe) secondary structures for that length. The map is established implicitly by a ....

.... in the set of empirical energy parameters or the thermodynamic level of description (one mfe structure versus the Boltzmann ensemble for a given sequence) 2 [22] These statistical properties, in particular neutrality, have led to a mathematical model based on percolation in random graphs [17]. In recent work we began to link the properties of the folding map with features observed in evolutionary dynamics. Model RNA populations in a ow reactor were subject to selection for a prespeci ed target structure. In particular, we found di usion of the population on a neutral network, and ....

C. Reidys, P. F. Stadler, and P. Schuster. Generic properties of combinatory maps - Neutral networks of RNA secondary structures. Bull. Math. Biol., 59:339-397, 1997.

....provides convincing evidence that RNA landscapes are as simple as they could possibly be for evolutionary adaptation. The consequences for evolutionary optimization, the early stages of life, and molecular biotechnology are immediate. Based on these findings a random graph theory was developed [25] that explains the structure of neutral networks in terms of a single parameter: the frequency of neutral mutations. The predictions of this 1 Gr uner et al. Analysis of RNA Sequence Structure Maps theory, among them connectedness and density of the neutral networks cannot be verified by ....

....structure ffl How is f Gamma1 (s) embedded in sequence space Many of these questions have been at least partially answered in previous papers describing non exhaustive computer simulations [5, 8, 6, 30, 31, 15] as mentioned in the introduction. Based on these results Reidys and coworkers [25] have constructed a random graph model for the preimage of a given secondary structure s which basically depends on a single parameter, namely the (average) fraction of neutral neighbors of the sequences folding into s, see the section on neutral networks below. The exhaustive enumeration data ....

[Article contains additional citation context not shown here]

C. Reidys, P. Schuster, and P. F. Stadler. Generic properties of combinatory maps --- neutral networks of rna secondary structures. Bull. Math. Biol., 1995. submitted; SFI-preprint 95-07-058.

No context found.

Reidys, C., Stadler, P., & Schuster, P. (1997). Generic properties of combinatory maps: neutral networks of rna secondary structures. Bulletin of Mathematical Biology, 59(2), 339--397.

No context found.

Reidys C, Stadler PF, Schuster P. 1997. Generic properties of combinatory maps: neutral networks of RNA secondary structures. Bull Math Biol 59:339--397.

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