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Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes

Tomoaki Imoto1, *, Grace S. Shieh2, *, Kunio Shimizu3

1 School of Management and Information, University of Shizuoka, Shizuoka, Japan.
2 Institute of Statistical Science, Academia Sinica, Taipei, Taiwan.
3 School of Statistical Thinking, The Institute of Statistical Mathematics, Tokyo, Japan.

* Corresponding Authors: Tomoaki Imoto. Email: ;
  Grace S. Shieh. Email: .

(This article belongs to this Special Issue: Data Science and Modeling in Biology, Health, and Medicine)

Computer Modeling in Engineering & Sciences 2020, 123(3), 1131-1149.


For structural comparisons of paired prokaryotic genomes, an important topic in synthetic and evolutionary biology, the locations of shared orthologous genes (henceforth orthologs) are observed as binned data. This and other data, e.g., wind directions recorded at monitoring sites and intensive care unit arrival times on the 24-hour clock, are counted in binned circular arcs, thus modeling them by discrete circular distributions (DCDs) is required. We propose a novel method to construct a DCD from a base continuous circular distribution (CCD). The probability mass function is defined to take the normalized values of the probability density function at some pre-fixed equidistant points on the circle. Five families of constructed DCDs which have normalizing constants in closed form are presented. Simulation studies show that DCDs outperform the corresponding CCDs in modeling grouped (discrete) circular data, and minimum chi-square estimation outperforms maximum likelihood estimation for parameters. We apply the constructed DCDs, invariant wrapped Poisson and wrapped discrete skew Laplace to compare the structures of paired bacterial genomes. Specifically, discrete four-parameter wrapped Cauchy (nonnegative trigonometric sums) distribution models multi-modal shared orthologs in Clostridium (Sulfolobus) better than the others considered, in terms of AIC and Freedman’s goodness-of-fit test. The result that different DCDs fit the shared orthologs is consistent with the fact they belong to two kingdoms. Nevertheless, these prokaryotes have a common favored site around 70° on the unit circle; this finding is important for building synthetic prokaryotic genomes in synthetic biology. These DCDs can also be applied to other binned circular data.


Cite This Article

Imoto, T., Shieh, G. S., Shimizu, K. (2020). Discrete Circular Distributions with Applications to Shared Orthologs of Paired Circular Genomes. CMES-Computer Modeling in Engineering & Sciences, 123(3), 1131–1149.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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