Author Correction: Informational laws of genome structures

Correction to: Scientific Reports 10.1038/srep28840, published online 29 June 2016

This Article contains errors in the Methods section, subheading ‘Mathematical Backgrounds’.

Formulation of Lemma 1 is incorrect:

“Given a genome $\mathbb{G}$ of length n, if $k=mrl(\mathbb{G})+1$, then $E_k(\mathbb{G})$ is the maximum value that $E_k$ can reach in the class of all possible genomes of length n”.

should read:

“Given a genome $\mathbb{G}$ of length n, if $k=mrl(\mathbb{G})+1$, then $E_k(G)$ is the maximum in the class of values $\{E_h(G)|n\ge h\ge k\}$”.

The corresponding proof:

“The minimum value of k such that all k-mers are hapaxes of $\mathbb{G}$ is $mrl(\mathbb{G})+1$. Therefore, if $k=mrl(\mathbb{G})+1$, then $E_k(\mathbb{G})$ is maximum, according to the entropy Equipartition Property, because we have the maximum number of words occurring once in $\mathbb{G}$, and all these words have the same probability of occurring in $\mathbb{G}$”.

should read:

“For $k=mrl+1$ empirical entropy $E_k(G)$ is equal to ${log}_2(n-k+1)$, in fact $(n-k+1)$ is the number of distinct k-mers in $\mathbb{G}$. The same expression holds for any $h\le n$ and $h\ge k$, because string longer than k are hapaxes too. But, if $h>k$, then ${log}_2(n-k+1)>{log}_2(n-h+1)$, therefore $E_k(G)=max\{E_h(G)|n\ge h\ge k\}$”.

The proof of Lemma 2 is incorrect for the lack of an explicit characterization of “random genomes”. Here a correct proof is given:

First at all, a random genome of length n is obtained by a random process of generation where, at each step, one of four possible genome symbols is generated with probability 1/4. Let $k=mrl+1$. According to the theory of de Brujin sequences, it is possible to arrange all $4^k$ possible k-mers in a circular sequence $\alpha$ (the last symbol of $\alpha$ is followed by the first symbol of $\alpha$) where each k-mer occurs exactly once. Of course, any contiguous portion long n of $\alpha$ contains $(n-k+1)$ consecutive k-mers and corresponds to a random genome of length n (shortly, a n-genome). In fact, all symbols of $\alpha$ are equiprobable, and this homogeneity holds along all positions of $\alpha$, in the sense that, going forward (circularly) a number of steps equal to the length of $\alpha$ another de Brujin sequence, with the same equiprobability property is obtained, Let us consider the disjoint n-genomes (with no common k-mer) concatenated in $\alpha$. Their number is $m=4^k/(n-k+1)$. But $\alpha$ is the shortest circular string arranging all k-mers, then, maximum statistical homogeneity (required by randomness) is reached when $m=(n-k+1)$, that is, when the probability that a k-mer has of occurring in one of the disjoint n-genomes of $\alpha$ is the same of occurring in one of the $(n-k+1)$ k-mer positions of a n-genome (a sort of scale-free equiprobability). This condition is expressed by equation (14) of the paper, from which equation $4^k={(n-k+1)}^2$ follows, corresponding to equation (15). Whence, equations (16), (17), (18) and the inequality (19) follow, from which the bounds given for $mrl+1$ derive.

Consequently, proposition 3’s opening sentence:

“In the class of genomes of length n, for every $k<n$, the following relation holds:”

should read:

“In the class of genomes of length n, for every $mrl+1\le k\le n$, the following relation holds:”

Finally, in Results, subheading ‘Information genomics laws’ Eq. 7 should be removed, because it follows from (8), being $LX>1$.