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On the Theory of Stochastic Replication and Evolution of Molecular Sequences

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Abstract

Following the hypothesis of M. Eigen the stochastic replication of molecular sequences is one of the basic processes in molecular biology. This paper is devoted to the formal analysis of the stochastic kinetics of replicating sequences and their evolutionary trees. For the microscopic descriptiuon of systems consisting of sequences of molecules, the metric space of sequences, the occupation number space and probability distributions in the latter are introduced. The complexity of a sequence is described by Kolmogorov's algorithmic entropy which is considered as a measure of the quantity of information contained in the sequence.The time evolution of sequence systems in the occupation number space is described by a dynamic semigroup corresponding to a master equation. The realizations of the stochastic process form an evolution tree which is analyzed by automata-theoretical methods. As an example a certain model of replicating sequences with 4 units is studied. As a criterion for evolutionary processes in sequence systems the increase of the mean Kolmogorov complexity in the system is proposed.
... WinGramm is used to generate and to analyze symbol sequences with the methods of subword entropies, grammatical complexities, and surrogate statistics. These are ways to study the information content, the complexity or the redundancy of any temporal data series in form of symbolic dynamics, of biomolecular sequences1234 of digital information carriers or of human writings[3]. For the details of these tools the user is referred to the literature listed at the end. ...
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1. General The program WinGramm ist freely distributable for non-commercial use only. It is intended for scientific and engineering applications by scientists, engineers, or students. WinGramm runs on Windows platforms, like Win95, Win98, WinNT 4.0, Win 2000. It does not need any special hardware configuration except a graphics display, but sufficient memory and processor speed may prove very recommendable for use with longer sequences. The authors cannot guarantee that the program is error-free. They do not take any warranty for possibly occuring problems caused by the use of the program or of its results. The use of the program is entirely at the risk of the user. On the other hand, hints on bugs or improvements of the program are always appreciated. To install WinGramm on your computer, copy the components of the installation package, - WinGra1.cab - WinGra2.cab - Setup.lst - Setup.exe into a suitable folder and start Setup.exe. WinGramm works with time series files and with sequence files. Time series should be files with a single figure - the current value - per line, either integer or floating point, without any headlines, separators etc. Text behind the figure (or additional figures) is ignored. Sequence files consist of a number of lines with a number of characters per line. This might be one character per line, or all characters in just one line, or a mixture like in ordinary ASCI text files. Each character found is treated as symbol of the sequence, with the following exceptions. The characters linefeed LF = and carriage return CR = are expected to separate the lines, they are ignored when reading. Note that there are no spaces inserted automatically in case you study usual text files. You should completely refrain from using the other control characters to , and , as symbols of your sequence. However, "space" = is a permitted symbol. Briefly, the list of the 223 usable characters to is
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Based on Chomsky's hierarchy of phrase structure grammars and on the complexity concept of Kolmogorov and Chaitin, an approach to the syntactic analysis of biopolymers is developed. Context-free and context-sensitive grammars are applied to the description of polypeptides and polynucleotides. The length of the minimal grammar program generating the sequence is used as a measure of its complexity, and the number of rules (derivational length) as a measure of the grammar structure. 25 DNA, RNA, and protein sequences are analyzed. It is shown that the derivational length has a tendency to increase along phylogenetic pathways. A proposal is made to connect the replication rates in the Eigen-Schuster theory with the concepts developed here.
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We define a new type of formal grammars where the derivation process is regulated by a certain function which evaluates the words. These grammars can be regarded as a model for the molecular replication process with selective character. We locate the associated family of languages in the Chomsky hierarchy, prove some closure properties, and solve some decision problems which are of interest in formal language theory and in biophysics.
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Evolutionary processes are described as stochastic motions in a genotype space (set of sequences with a Hamming distance) and a phenotype space (vector space of phenotypic properties). Real value functions are introduced which form a landscape over these spaces; smoothness postulates are formulated. Evolution is considered as a kind of hill climbing on these adaptive landscapes. A rather simple diffusion approximation for the phenotypic processes is proposed which leads to similar mathematical problems as the Schrödinger equation for disordered potential distributions.
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A representation of the genetic code as a six-dimensional Boolean hypercube is described. This structure is the result of the hierarchical order of the interaction energies of the bases in codon-anticodon recognition. In this paper it is applied to study molecular evolution in vivo and in vitro. In the first case we compared aligned positions in homologous protein sequences and found two different behaviors: (a) There are sites in which the different amino acids may be explained by one or two 'attractor nodes' (coding for the dominating amino acid(s)) and their one-bit neighbors in the codon hypercube; and (b) There are sites in which the amino acids correspond to codons located in closed paths in the hypercube. In the second case we studied the 'Sexual PCR'1 experiment described by Stemmer [Stemmer (1994)] and found that the success of this combination of usual PCR and recombination is in part due to the Gray code structure of the genetic code.
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