Word frequency distribution under restriction avoidance

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Restriction enzymes of bacteria cut the unmodified recognition sites of DNA, thereby protect bacteria from their parasites. This potentially imposes strong selection on phage genome sequences.. To answer the evolution of phage genome under the selection by bacteria restriction enzymes, a simple model for the evolution of binary sequences is proposed. The model shows that not only the restricted sequence itself but the relative abundance of associated subsequences (words) in the genome will be largely deviated from the random expectations. The word frequency distribution sensitively depends on the type of restricted sequence. If the restricted sequence is of m type, where x is either 0 or 1, the frequency of a 3-letter word is largely explained by its Hamming distance from the restricted sequence -the farther is the Hamming distance from restriction sequence, the more is the abundance of that subsequence. However, quite unexpected word frequency distribution arises if the restricted sequence is of xxy or xyx type. The abundance of words is largely influenced by the vulnerability to restriction of partially overlapped adjacent words. For example, when 001 is the restriction sequence, the word 100, which is two steps distant from the restriction sequence, becomes quite rare in the genome. Such distribution of words under restriction is well explained by counting the relative abundance of words in the feasible sequences, which are the sequences without any restriction site. Simple formula for the word distribution in the feasible sequences are obtained.