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  • 1
    Electronic Resource
    Electronic Resource
    Dynamics of small autocatalytic reaction networks—II. Replication, mutation and catalysis (1995)
    Springer
    Bulletin of mathematical biology 57 (1995), S. 21-61 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mutation is introduced into autocatalytic reaction networks. The differential equations obtained are neither of repliator-type nor can they be transformed straightway into a linear equation. Examples of low dimensional dynamical systems —n=2, 3 and 4 — are discussed and complete qualitative analysis is presented. Error thresholds known from simple replication-mutation kinetics with frequency independent replication rates occur here as well. Instead of cooperative transitions or higher order phase transitions the thresholds appear here as supercritical or subcritical bifurcations being analogous to first-order phase transitions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458315
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  • 2
    Electronic Resource
    Electronic Resource
    RNA structures with pseudo-knots: Graph-theoretical, combinatorial, and statistical properties (1999)
    Springer
    Bulletin of mathematical biology 61 (1999), S. 437-467 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The secondary structures of nucleic acids form a particularly important class of contact structures. Many important RNA molecules, however, contain pseudo-knots, a structural feature that is excluded explicitly from the conventional definition of secondary structures. We propose here a generalization of secondary structures incorporating ‘non-nested’ pseudo-knots, which we call bi-secondary structures, and discuss measures for the complexity of more general contact structures based on their graph-theoretical properties. Bi-secondary structures are planar trivalent graphs that are characterized by special embedding properties. We derive exact upper bounds on their number (as a function of the chain length n) implying that there are fewer different structures than sequences. Computational results show that the number of bi-secondary structures grows approximately like 2.35n. Numerical studies based on kinetic folding and a simple extension of the standard energy model show that the global features of the sequence-structure map of RNA do not change when pseudo-knots are introduced into the secondary structure picture. We find a large fraction of neutral mutations and, in particular, networks of sequences that fold into the same shape. These neutral networks percolate through the entire sequence space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1998.0085
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  • 3
    Electronic Resource
    Electronic Resource
    Autocatalytic networks with translation (1996)
    Springer
    Bulletin of mathematical biology 58 (1996), S. 877-905 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We consider the kinetics of an autocatalytic reaction network in which replication and catalytic actions are separated by a translation step. We find that the behaviour of such a system is closely related to second-order replicator equations, which describe the kinetics of autocatalytic reaction networks in which the replicators act also as catalysts. In fact, the qualitative dynamics seems to be described almost entirely be the second-order reaction rates of the replication step. For two species we recover the qualitative dynamics of the replicator equations. Larger networks show some deviations, however. A hypercyclic system consisting of three interacting species can converge toward a stable limit cycle in contrast to the replicator equation case. A singular perturbation analysis shows that the replication-translation system reduces to a second-order replicator equation if translation is fast. The influence of mutations on replication-translation networks is also very similar to the behavior of selection-mutation equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459488
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  • 4
    Electronic Resource
    Electronic Resource
    Selection dynamics in autocatalytic systems: Templates replicating through binary ligation (1998)
    Springer
    Bulletin of mathematical biology 60 (1998), S. 1073-1098 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The theory of autocatalytic binary ligation is reviewed within the context of a consistently applied Michaelis-Menten quasi-steady-state approximation to obtain explicit analytical results describing time-course data from experiments. A detailed protocol for the step-wise elucidation of a minimal set of experimental parameters is outlined. The kinetic equations are then generalized to cases of self-and cross-catalysis among an arbitrary number of different templates and applied to experiments involving just two templates. Depending on the values of various kinetic parameters such systems can display exclusionary Darwinian selection corresponding to an exponential growth law, selective coexistence or coexistence of all species characteristic of a parabolic growth law; the intermediate behaviour arises as a property of the full mechanism analysed here. Our results are applicable to the classical case of self-replicating nucleic acids and their analogues as well as to newly discovered self-replicating peptides.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/S0092-8240(98)90003-9
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  • 5
    Electronic Resource
    Electronic Resource
    Population dependent Fourier decomposition of fitness landscapes over recombination spaces: Evolvability of complex characters (2000)
    Springer
    Bulletin of mathematical biology 62 (2000), S. 399-428 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes. Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p 〈 n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal. Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1999.0167
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  • 6
    Electronic Resource
    Electronic Resource
    Dynamics of autocatalytic replicator networks based on higher-order ligation reactions (2000)
    Springer
    Bulletin of mathematical biology 62 (2000), S. 1061-1086 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A class of autocatalytic reaction networks based on template-dependent ligation and higher-order catalysis is analysed. Apart from an irreversible ligation reaction we consider only reversible aggregation steps that provide a realistic description of molecular recognition. The overall dynamics can be understood by means of replicator equations with highly non-linear interaction functions. The dynamics depends crucially on the total concentration c 0 of replicating material. For small c 0, in the hyperbolic growth regime, we recover the familiar dynamics of second-order replicator equations with its wealth of complex dynamics ranging from multi-stability to periodic and strange attractors as well as to heteroclinic orbits. For large c 0, in the parabolic growth regime, product inhibition becomes dominating and we observe a single globally stable equilibrium tantamount to permanent coexistence. In an intermediate parameter range we sometimes observe a behavior that is reminiscent of ’survival of the fittest’. Independently replicating species (Schlögl’s model) and the hypercycle are discussed in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.2000.0194
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  • 7
    Electronic Resource
    Electronic Resource
    Small autocatalytic reaction networks—III. Monotone growth functions (1991)
    Springer
    Bulletin of mathematical biology 53 (1991), S. 469-485 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The classification of the dynamical behaviour of first order replicator equations is extended to models with monotonical growth rates. It is shown that for two species there is a general classification independent of the particular form of the growth function. For three species a common dynamical behaviour for all power laws can be found and the existence of limit cycles is disproved. For more general growth functions, however, limit cycles may occur.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460729
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  • 8
    Electronic Resource
    Electronic Resource
    Dynamics of small autocatalytic reaction networks—I. bifurcations, permanence and exclusion (1990)
    Springer
    Bulletin of mathematical biology 52 (1990), S. 485-508 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Catalysis in replication networks has become an important issue in biophysics and other areas of biology. Examples are RNA catalysis, idiotype recognition in the immune response and dynamical models of Maynard-Smith games in sociobiology. Chemical reaction networks describing catalysed, template-induced reproduction of three species are analysed in full generality. The nine-dimensional parameter space is reduced to three relevant angular coordinates which determine completely the phase portraits (PPs) and the bifurcation patterns. All cases are classified and all generic as well as most of the nongeneric transitions are listed and described.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462263
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  • 9
    Electronic Resource
    Electronic Resource
    Generic properties of combinatory maps: Neutral networks of RNA secondary structures (1997)
    Springer
    Bulletin of mathematical biology 59 (1997), S. 339-397 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Random graph theory is used to model and analyse the relationship between sequences and secondary structures of RNA molecules, which are understood as mappings from sequence space into shape space. These maps are non-invertible since there are always many orders of magnitude more sequences than structures. Sequences folding into identical structures formneutral networks. A neutral network is embedded in the set of sequences that arecompatible with the given structure. Networks are modeled as graphs and constructed by random choice of vertices from the space of compatible sequences. The theory characterizes neutral networks by the mean fraction of neutral neighbors (λ). The networks are connected and percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value (λ〉λ*). Below threshold (λ〈λ*), the networks are partitioned into a largest “giant” component and several smaller components. Structure are classified as “common” or “rare” according to the sizes of their pre-images, i.e. according to the fractions of sequences folding into them. The neutral networks of any pair of two different common structures almost touch each other, and, as expressed by the conjecture ofshape space covering sequences folding into almost all common structures, can be found in a small ball of an arbitrary location in sequence space. The results from random graph theory are compared to data obtained by folding large samples of RNA sequences. Differences are explained in terms of specific features of RNA molecular structures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462007
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  • 10
    Electronic Resource
    Electronic Resource
    Statistics of landscapes based on free energies, replication and degradation rate constants of RNA secondary structures (1991)
    Springer
    Monatshefte für Chemie 122 (1991), S. 795-819 
    Details
    ISSN: 1434-4475
    Keywords: RNA secondary structures ; RNA free energies ; Value landscapes ; Autocorrelation functions ; Correlation lengths
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: Zusammenfassung RNA-Sekundärstrukturen werden aus den Primärsequenzen mit Hilfe eines Computeralgorithmus berechnet, welcher einem Kriterium minimaler freier Energien folgt. Freie Energien, Replikations- oder Abbaugeschwindigkeitskonstanten werden aus den Sekundärstrukturen berechnet. Man kann daher diese Eigenschaften als komplizierte Funktionen der Sequenzen auffassen, deren Zahlenwerte durch Vermittlung der Sekundärstrukturen erhalten werden. Diese Funktionen induzieren hochkomplexe Bewertungslandschaften im Raum der Sequenzen. Die Landschaften werden mit Hilfe von Irrflugtechniken analysiert. Im einzelnen werden Autokorrelationsfunktionen und Korrelationslängen berechnet. Die freien Energie-Landschaften sind vom AR(1) Typ. Die von den Reaktionsgeschwindigkeitskonstanten abgeleiteten Landschaften stellten sich hingegen als komplexer heraus. Zusätzlich werden die Bewertungslandschaften auch noch mit Hilfe vonGradient undAdaptive Walks untersucht, um mehr Einblick in ihre komplexe Struktur zu gewinnen.
    Notes: Summary RNA secondary structures are computed from primary sequences by means of a folding algorithm which uses a minimum free energy criterion. Free energies as well as replication and degradation rate constants are derived from secondary structures. These properties can be understood as highly sophisticated functions of the individual sequences whose values are mediated by the secondary structures. Such functions induce complex value landscapes on the space of sequences. The landscapes are analysed by random walk techniques, in particular autocorrelation functions and correlation lengths are computed. Free energy landscapes were found to be of AR(1) type. The rate constant landscapes, however, turned out to be more complex. In addition, gradient and adaptive walks are performed in order to get more insight into the complex structure of the landscapes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00815919
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