Christian M. Reidys

Analysis of RNA sequence structure maps by exhaustive enumeration I. Neutral networks

SummaryGlobal relations between RNA sequences and secondary structures are understood as mappings from sequence space into shape space. These mappings are investigated by exhaustive folding of allGC andAU sequences with chain lengths up to 30. The computed structural data are evaluated through exhaustive enumeration and used as an exact reference for testing analytical results derived from mathematical models and sampling based on statistical methods. Several new concepts of RNA sequence to secondary structure mappings are investigated, among them that ofneutral networks (being sets of sequences folding into the same structure). Exhaustive enumeration allows to test several previously suggested relations: the number of (minimum free energy) secondary structures as a function of the chain length as well as the frequency distribution of structures at constant chain length (commonly resulting in generalized forms ofZipfs law).ZusammenfassungDie globalen Benziehungen zwischen RNA-Sequenzen und Sekundärstrukturen werden als Abbildungen aus einem Raum aller Sequenzen in einen Raum aller Strukturen aufgefaßt. Diese Abbildungen werden durch Falten aller binären Sequenzen desGC-undAU-Alphabets mit Kettenlängen bis zun=30 untersucht. Die berechneten Strukturdaten werden durch vollständiges Abzählen ausgewertet und als eine exakte Referenz zum Überprüfen analytischer Resultate aus mathematischen Modellen sowie zum Testen statistisch erhobener Proben verwendet. Einige neuartige Konzepte zur Beschreibung der Beziehungen zwischen Sequenzen und Strukturen werden eingehend untersucht, unter ihnen der Begriff derneutralen Netzwerke. Ein neutrales Netzwerk besteht aus allen Sequenzen, die eine bestimmte Struktur ausbilden. Vollständiges Abzählen ermöglicht beispielsweise die Bestimmung aller Strukturen minimaler freier Energie in Abhängigkeit von der Kettenlänge ebenso wie die Bestimmung der Häufigkeitsverteilungen der Strukturen bei konstanten Kettenlängen. Die letzteren folgen einer verallgemeinerten FormZipfschen Gesetzes.

Analysis of RNA sequence structure maps by exhaustive enumeration II. Structures of neutral networks and shape space covering

SummaryThe relations between RNA sequences and secondary structures are investigated by exhaustive folding of allGC andAU sequences with chain lengths up to 30. The technique oftries is used for economic data storage and fast retrieval of information. The computed structural data are evaluated through exhaustive enumeration and used as an exact reference for testing analytical results derived from mathematical models and sampling based on statistical methods. Several new concepts of RNA sequence to secondary structure mappings are investigated, among them the structure ofneutral networks (being sets of RNA sequences folding into the same structure),percolation of sequence space by neutral networks, and the principle ofshape space covering. The data of exhaustive enumeration are compared to the analytical results of arandom graph model that reveals the generic properties of sequence to structure mappings based on some base pairing logic. The differences between the numerical and the analytical results are interpreted in terms of specific biophysical properties of RNA molecules.ZusammenfassungDie Beziehungen zwischen RNA-Sequenzen und ihren Sekundärstrukturen werden durch vollständiges Falten allerGC- undAU-Sequenzen mit Kettenlängen bis zun=30 untersucht. Die aus der Informatik bekannte Technik derTries wird zur ökonomischen Datenspeicherung und für rasches Retrieval der gespeicherten Information angewendet. Die berechneten Strukturdaten werden durch vollständiges Abzählen ausgewertet. Sie dienen unter anderem als eine exakte Referenz zum Testen analytischer Resultate aus mathematischen Modellen sowie zur Überprüfung der Ergebnisse statistischer Probennahmen. Verschiedene neuartige Konzepte zur Behandlung der Zusammenhänge zwischen RNA-Sequenzen und Sekundärstrukturen wurden anhand der gewonnenen Daten eingehend untersucht. Unter ihnen befinden sich die Struktur derneutralen Netzwerke (die Gesamtheit der RNA-Sequenzen, die eine bestimmte Struktur ausbilden), diePerkolation des Sequenzraumes durch neutrale Netzwerke sowie das Prinzip derErfassung des Strukturraumes durch einen kleinen Ausschnitt des Sequenzraumes. Die durch vollständiges Abzählen erhaltenen Daten werden mit den analytischen Ergebnissen eines auf der Theorie der Zufallsgraphen aufbauenden Modells verglichen. Dieses Modell gibt die generischen Eigenschaften von Sequenz-Struktur-Relationen wieder, welche lediglich aus der Existenz einer Paarungslogik resultieren. Differenzen zwischen den numerischen und den analytischen Resultaten können als Konsequenzen der spezifischen biophysikalischen Eigenschaften von RNA-Molekülen interpretiert werden.

Neutrality in fitness landscapes

The interplay of ruggedness and neutrality in fitness landscapes plays an important role in explaining the dynamics of evolutionary adaptation. While various measures of ruggedness (correlation functions, adaptive walks, or the density of local optima) are reasonably well understood, and models for constructing landscapes with a desired degree of ruggedness are readily available, very little is known about neutrality. We introduce the notion of additive random landscapes as a framework for tuning both neutrality and ruggedness at once, and we develop a formalism that allows the explicit computation of the most salient parameters that are associated with neutrality in landscapes of this type.

Discrete, sequential dynamical systems

Abstract We study a class of discrete dynamical systems that consists of the following data: (a) a finite loop-free graph Y with vertex set {1, …, n} where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F i, Y : F 2 n → F 2 n ) i and (c) a permutation π∈Sn. The function F i, Y updates the state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F i, Y are applied. By composing the functions F i, Y in the order given by π we obtain the dynamical system [ F Y , π]=∏ i=1 n F π(i),Y : F 2 n → F 2 n , which we refer to as a sequential dynamical system (SDS). Among various basic results on SDS we will study their invertibility and analyze the set |{[ F Y , π] | π∈S n }| for fixed Y and (F i, Y ) i . Finally, we give an estimate for the number of non-isomorphic digraphs Γ[ F Y , π] (having vertex sets F 2 n and directed edges {(x, [ F Y , π](x)) | x∈ F 2 n } ) for a fixed graph Y and a fixed multi-set (F i, Y ) i .

Bio-Molecular Shapes and Algebraic Structures

Shapes of biological macromolecules--RNA, DNA, and proteins--can be represented by abstract algebraic structures provided that a suitably coarse resolution is chosen. These abstract structures, for instance partially ordered sets and permutation groups, can be used for deriving new metric distances between bimolecular shapes and for proving surprising theorems on sequence-structure relations.

Asymptotic Enumeration of RNA Structures with Pseudoknots

AbstractIn 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 asymptotic expansion for the numbers of k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures,

Inverse folding of RNA pseudoknot structures

{\mathsf{S}}_{k}(n)

Folding 3-noncrossing RNA pseudoknot structures.

, derived in Bull. Math. Biol. (2008), where k−1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function

Random k-noncrossing RNA structures

\sum_{n\ge 0}{\mathsf{S}}_{k}(n)z^{n}

ON ASYNCHRONOUS CELLULAR AUTOMATA

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 3-noncrossing RNA structures. Our main result is the derivation of the formula