This is an investigation of methods for finding short motifs that only occur in a fraction of the input sequences. Unlike local search techniques that may not reach a global optimum, the method proposed here is guaranteed to produce the motifs with greatest z-scores. This method is illustrated for the Ribosome Binding Site Problem, which is to identify the short mRNA 5' untranslated sequence that is recognized by the ribosome during initiation of protein synthesis. Experiments were performed to solve this problem for each of fourteen sequenced prokaryotes, by applying the method to the full complement of genes from each. One of the interesting results of this experimentation is evidence that the recognized sequence of the thermophilic archaea A. fulgidus, M. jannaschii, M. thermoautotrophicum, and P. horikoshii may be somewhat different than the well known Shine-Dalgarno sequence.

/pdf/an-exact-method-for-finding-short-motifs-in-sequences-with-27l972k833.pdf

An Exact Method for Finding Short Motifs in Sequences, with Application to the Ribosome Binding Site Problem

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2.m
conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.

/pdf/sooner-and-later-waiting-time-problems-for-success-and-2h0zu07qhq.pdf

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

In this article, the probability and moment generating functions of the number of appearances of a pattern e in a sequence of discrete random variables (repeated trials) are expressed in terms of the generating function of the waiting time for the r-th occurrence of e, The special case of delayed recurrent events is also examined in some detail. Finally, the general theory is employed for a systematic investigation of success runs enumeration problems in a sequence of binary outcomes arising from a first-order Markov chain.

Waiting Times and Number of Appearances of Events in a Sequence of Discrete Random Variables

Scan statistics are commonly used in biology, medicine, engineering and other fields where interest is in the probability of observing clusters of events in a window at an unknown location Due to the dependent nature of the number of events in a large number of overlapping window locations, even approximate solutions for the simplest scan statistics may require elaborate calculations We propose a new martingale method which allows one to approximate the distribution for a wide variety of scan statistics, including some for which analytical results are computationally infeasible

/pdf/a-martingale-approach-to-scan-statistics-4td3my4797.pdf

A martingale approach to scan statistics

Consider a given pattern H and a random text T generated randomly according to the Bernoulli model. We study the frequency of approximate occurrences of the pattern H in a random text when overlapping copies of the approximate pattern are counted separately. We provide exact and asymptotic formulae for mean, variance and probability of occurrence as well as asymptotic results including the central limit theorem and large deviations. Our approach is combinatorial: we first construct some language expressions that characterize pattern occurrences which are translated into generating functions, and finally we use analytical methods to extract asymptotic behaviours of the pattern frequency. Applications of these results include molecular biology, source coding, synchronization, wireless communications, approximate pattern matching, games, and stock market analysis. These findings are of particular interest to information theory (e.g., second-order properties of the relative frequency), and molecular biology problems (e.g., finding patterns with unexpected high or low frequencies, and gene recognition).

/pdf/on-the-approximate-pattern-occurrences-in-a-text-24avjvlccm.pdf

On the approximate pattern occurrences in a text

A sequence of Markov dependent experiments is performed, each one of them producing a letter from a given alphabet. We study the following question: How many trials does it take to observe a member of a specified finite set of patterns of letters for the first time?

The Occurrence of Sequence Patterns in Repeated Dependent Experiments

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The Occurrence of Sequence Patterns in Repeated Dependent Experiments