White noise functional integral for exponentially decaying memory: nucleotide distribution in bacterial genomes

Abstract

We utilize a stochastic functional integral approach that forms a natural framework for analyzing ubiquitous complex sequences of fluctuations with underlying non-Markovian stochastic process beyond fractional Brownian motion. We demonstrate how Hida white noise calculus, guided by mean square deviation (MSD) analysis of empirical data, allows derivation of single nucleotide occurrence probability distributions for whole genomes of four significant species of bacteria: (a) freshwater cyanobacteria Synechococcus elongatus PCC7942, 2.7 Mbp, (b) marine cyanobacteria Prochlorococcus marinus subsp. marinus str. CCMP1375, 1.8 Mbp, (c) pathogenic bacteria Staphylococcus aureus subsp. aureus NCTC 8325, 2.8 Mbp, and (d) Staphylococcus aureus ILRI Eymole1/1, 2.9 Mbp. Here, the stochastic variable is chosen to represent separation distances between succeeding identical single nucleotides where distance is defined as the number of steps through intervening bases. The stochastic parameter set takes values of nucleotide occurrence count along the genome length. The probability density function (PDF) is derived in closed form for the associated stochastic process with exponentially damped memory kernel, and is shown to satisfy a modified diffusion equation with a parameter-dependent diffusion coefficient. The PDF yields an analytical result for MSDs that match empirical plots, showing a rising nonlinear curve that flattens to a plateau starting close to 1 kb, similar to restricted diffusion. The plots exhibit compliance with Chargaff’s second parity rule for nucleotides. The same PDF describes occurrences of single nucleotides adenine, guanine, cytosine, and thymine for all four bacterial genomes considered.