Asymptotic results for the multiple scan statistic

The contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

Nonrandom clusters of palindromes in herpesvirus genomes

Palindromes are symmetrical words of DNA in the sense that they read exactly the same as their reverse complementary sequences. Representing the occurrences of palindromes in a DNA molecule as points on the unit interval, the scan statistics can be used to identify regions of unusually high concentration of palindromes. These regions have been associated with the replication origins on a few herpesviruses in previous studies. However, the use of scan statistics requires the assumption that the points representing the palindromes are independently and uniformly distributed on the unit interval. In this paper, we provide a mathematical basis for this assumption by showing that in randomly generated DNA sequences, the occurrences of palindromes can be approximated by a Poisson process. An easily computable upper bound on the Wasserstein distance between the palindrome process and the Poisson process is obtained. This bound is then used as a guide to choose an optimal palindrome length in the analysis of a collection of 16 herpesvirus genomes. Regions harboring significant palindrome clusters are identified and compared to known locations of replication origins. This analysis brings out a few interesting extensions of the scan statistics that can help formulate an algorithm for more accurate prediction of replication origins.

Nonoverlapping clusters: approximate distribution and application to molecular biology

An approach is developed for the screening of genomic sequence data to identify gene regulatory regions. This approach is based on deciding if putative transcription factor binding sites are clustered together to a greater extent than one would expect by chance. Given n events occurring on an interval of width L (L base pairs), an r:w cluster is defined as r + 1 consecutive events all contained within a window of length wL. Accurate and easily computable approximations are derived for the distribution of the number of nonoverlapping r:w clusters under the model that the positions of the n events have a uniform distribution. Simulations demonstrate that these approximations have greater accuracy than existing methods. The approximation is applied to detect erythroid-specific regulatory regions in genomic DNA sequences, first in an artificial case where r is specified a priori and then as part of an exploratory approach.