Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarize and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the expected type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.

We present a comprehensive review on probabilistic arithmetic automata (PAAs), a general model to describe chains of operations whose operands depend on chance, along with two algorithms to numerically compute the distribution of the results of such probabilistic calculations. PAAs provide a unifying framework to approach many problems arising in computational biology and elsewhere. We present five different applications, namely 1) pattern matching statistics on random texts, including the computation of the distribution of occurrence counts, waiting times, and clump sizes under hidden Markov background models; 2) exact analysis of window-based pattern matching algorithms; 3) sensitivity of filtration seeds used to detect candidate sequence alignments; 4) length and mass statistics of peptide fragments resulting from enzymatic cleavage reactions; and 5) read length statistics of 454 and IonTorrent sequencing reads. The diversity of these applications indicates the flexibility and unifying character of the presented framework. While the construction of a PAA depends on the particular application, we single out a frequently applicable construction method: We introduce deterministic arithmetic automata (DAAs) to model deterministic calculations on sequences, and demonstrate how to construct a PAA from a given DAA and a finite-memory random text model. This procedure is used for all five discussed applications and greatly simplifies the construction of PAAs. Implementations are available as part of the MoSDi package. Its application programming interface facilitates the rapid development of new applications based on the PAA framework.

BackgroundMolecular database search tools need statistical models to assess the significance for the resulting hits. In the classical approach one asks the question how probable a certain score is observed by pure chance. Asymptotic theories for such questions are available for two random i.i.d. sequences. Some effort had been made to include effects of finite sequence lengths and to account for specific compositions of the sequences. In many applications, such as a large-scale database homology search for transmembrane proteins, these models are not the most appropriate ones. Search sensitivity and specificity benefit from position-dependent scoring schemes or use of Hidden Markov Models. Additional, one may wish to go beyond the assumption that the sequences are i.i.d. Despite their practical importance, the statistical properties of these settings have not been well investigated yet.ResultsIn this paper, we discuss an efficient and general method to compute the score distribution to any desired accuracy. The general approach may be applied to different sequence models and and various similarity measures that satisfy a few weak assumptions. We have access to the low-probability region ("tail") of the distribution where scores are larger than expected by pure chance and therefore relevant for practical applications. Our method uses recent ideas from rare-event simulations, combining Markov chain Monte Carlo simulations with importance sampling and generalized ensembles. We present results for the score statistics of fixed and random queries against random sequences. In a second step, we extend the approach to a model of transmembrane proteins, which can hardly be described as i.i.d. sequences. For this case, we compare the statistical properties of a fixed query model as well as a hidden Markov sequence model in connection with a position based scoring scheme against the classical approach.ConclusionsThe results illustrate that the sensitivity and specificity strongly depend on the underlying scoring and sequence model. A specific ROC analysis for the case of transmembrane proteins supports our observation.

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We present probabilistic arithmetic automata (PAAs), a general model to describe chains of operations whose operands depend on chance, along with two different algorithms to exactly calculate the distribution of the results obtained by such probabilistic calculations. PAAs provide a unifying framework to approach many problems arising in computational biology and elsewhere. Here, we present five different applications, namely (1) pattern matching statistics on random texts, including the computation of the distribution of occurrence counts, waiting time and clump size under HMM background models; (2) exact analysis of window-based pattern matching algorithms; (3) sensitivity of filtration seeds used to detect candidate sequence alignments; (4) length and mass statistics of peptide fragments resulting from enzymatic cleavage reactions; and (5) read length statistics of 454 sequencing reads. The diversity of these applications indicates the flexibility and unifying character of the presented framework.While the construction of a PAA depends on the particular application, we single out a frequently applicable construction method for pattern statistics: We introduce deterministic arithmetic automata (DAAs) to model deterministic calculations on sequences, and demonstrate how to construct a PAA from a given DAA and a finite-memory random text model. We show how to transform a finite automaton into a DAA and then into the corresponding PAA. * Parts of this article have been published in conference proceedings [29,47,23,48,50]. An extended version of one of these articles [50] has been submitted to a journal. A preprint is available on arXiv [49].

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