Latent Periodicities in Genome Sequences

Abstract

A novel approach is presented to the detection of periodicities in DNA sequences. A DNA sequence can be modelled as a nonstationary stochastic process that exhibits various statistical periodicities in different regions.The coding part of the DNA, for instance, exhibits statistical periodicity with period three. Such regions in DNA are modelled as generated from a collection of information sources (with an underlying probability distribution) in a cyclic manner, thus exhibiting cyclostationarity. The maximum likelihood estimates are developed for the distributions of the information sources and for the statistical period of the DNA sequence. Such sequences are further investigated for decomposition into constituent cyclostationary sources. Since the symbolic sources do not admit an algebraic structure, a composition of cyclostationary probabilistic sources is studied that models the point mutations in gene sequences. This composition is shown to give a rich mathematical structure on the collection of cyclostationary sources and allows a uniqueness theorem for the decomposition of statistically periodic symbolic sources.

Estimating Periodicities in Symbolic Sequences Using Sparse Modeling

“…From the figure, one can also observe that, as expected, the PECS L outperforms the PECS G when there is more than one periodicity present in the sequence. For the last simulated data experiment, we recreate a simulation experiment similar to the one that was used in [8], where a deterministic periodicity of 11 and 31 are present simultaneously in a signal generated from a 4 element set being uniformly distributed on the other indices. As can be seen in Figure 5, the PECS G estimator achieves almost 100 % success rate even before the method presented in [8] can start to be used, since it requires a minimum of 11 × 31 = 341 data points.…”

“…For the last simulated data experiment, we recreate a simulation experiment similar to the one that was used in [8], where a deterministic periodicity of 11 and 31 are present simultaneously in a signal generated from a 4 element set being uniformly distributed on the other indices. As can be seen in Figure 5, the PECS G estimator achieves almost 100 % success rate even before the method presented in [8] can start to be used, since it requires a minimum of 11 × 31 = 341 data points. Finally, we examine the performance of the PECS G estimator on measured genomic data, in the form of the gene C. elegans F56F11.4 [28].…”

“…, m}, where · denotes the rounding down operation. This means that if a periodicity m is present in a sequence, the sequence is clearly also periodic on the subharmonics i.e., for every mr:th symbol, for all natural numbers r [8]. To avoid ambiguity, we here refer to the period as the lowest possible such periodicity.…”

“…Comparing the likelihood above with (3), it can be seen that (11) [8], although there they defined a statistical periodicity, say k, to be present when all index set I(k, ), for = 1 . .…”

“…Thus, one may form a test to determine the hypothesis that a given sequence has a different distribution for the indices corresponding to I(m, ), i.e., that the PMF is formed using (8), against the null hypothesis that the entire sequence has the same categorical distribution, such that the PMF instead follows (3), i.e.,…”