Computational Biology and Chemistry 53 (2014) 15–25 Contents lists available at ScienceDirect Computational Biology and Chemistry journal homepage: www.elsevier.com/locate/compbiolchem Bacterial genomes lacking long-range correlations may not be modeled by low-order Markov chains: The role of mixing statistics and frame shift of neighboring genes Germinal Cocho a , Pedro Miramontes b,* , Ricardo Mansilla c , Wentian Li d,* a Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico 04510, DF, Mexico b Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 04510, DF, Mexico c Centro de Investigaciones Interdisciplinarias en Ciencias y Hamanidades, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico 04510, DF, Mexico d The Robert S. Boas Center for Genomics and Human Genetics, The Feinstein Institute for Medical Research, North Shore LIJ Health System, Manhasset, NY, USA article info Article history: Available online 30 August 2014 Keywords: Bacterial genomes Exponential correlation function Markov model Second largest eigenvalue Hexamer Periodicity of 10–11 bases Heterogeneity Codon positions abstract We examine the relationship between exponential correlation functions and Markov models in a bacterial genome in detail. Despite the well known fact that Markov models generate sequences with correla- tion function that decays exponentially, simply constructed Markov models based on nearest-neighbor dimer (first-order), trimer (second-order), up to hexamer (fifth-order), and treating the DNA sequence as being homogeneous all fail to predict the value of exponential decay rate. Even reading-frame-specific Markov models (both first- and fifth-order) could not explain the fact that the exponential decay is very slow. Starting with the in-phase coding-DNA-sequence (CDS), we investigated correlation within a fixed- codon-position subsequence, and in artificially constructed sequences by packing CDSs with out-of-phase spacers, as well as altering CDS length distribution by imposing an upper limit. From these targeted anal- yses, we conclude that the correlation in the bacterial genomic sequence is mainly due to a mixing of heterogeneous statistics at different codon positions, and the decay of correlation is due to the possible out-of-phase between neighboring CDSs. There are also small contributions to the correlation from bases at the same codon position, as well as by non-coding sequences. These show that the seemingly simple exponential correlation functions in bacterial genome hide a complexity in correlation structure which is not suitable for a modeling by Markov chain in a homogeneous sequence. Other results include: use of the (absolute value) second largest eigenvalue to represent the 16 correlation functions and the prediction of a 10–11 base periodicity from the hexamer frequencies. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Long-range correlations often refer to a power-law correlation function, as versus short-range correlations referring in exponen- tial correlation function. Many genomes, when a chromosome is treated as a sequence of symbols or numerical values, exhibit power-law long-range correlations (Li, 1997a; Buldyrev, 2006; Arneodo et al., 2011). More interestingly, the type of long-range correlations in genomes share similarity with the “1/f noise” time series (Li and Kaneko, 1992; Voss, 1992; Li et al., 1998; Li and Holste, 2005). Not all genomes exhibit power-law correlation functions, * Corresponding authors. E-mail addresses: pmv@ciencias.unam.mx (P. Miramontes), wtli2012@gmail.com (W. Li). however – the bacteria genomes tend to exhibit 1/f 2 spectra (Li, 1997b) and exponential correlation functions (Bernaola-Galván et al., 2002). There are many mathematical models of sequences with power- law correlations (Beran, 1994; Beran et al., 2014). Although there are attempts to propose a universal framework for all observed power-laws (Peterson et al., 2013), the mechanical model of any specific dataset with power-law distributions could be non- universal and not applicable to other datasets (Sornette, 2006). For example, many long-range correlations of complex genomes may be caused by large domains with differential base composi- tions, whose size follow a broad or even long-tailed distribution (Bernaola-Galván et al., 1996; Clay et al., 2001). The range of mathematical models of sequences with exponen- tial correlation function, on the other hand, is relatively narrow. Markov chains are almost always used as the generating model. http://dx.doi.org/10.1016/j.compbiolchem.2014.08.005 1476-9271/© 2014 Elsevier Ltd. All rights reserved.