Word frequency distribution under restriction avoidance

细菌的限制性内切酶切断DNA未修饰的识别位点，从而保护细菌免受寄生虫的侵害。这可能会对噬菌体基因组序列施加强烈的选择。为了回答噬菌体基因组在细菌限制性内切酶选择下的进化，提出了一个简单的二元序列进化模型。该模型表明，不仅限制序列本身，而且基因组中相关子序列（单词）的相对丰度也将在很大程度上偏离随机预期。词频分布敏感地依赖于受限序列的类型。如果限制序列为m型，其中x为0或1，则3个字母单词的频率在很大程度上由其与限制序列的汉明距离来解释——距离限制序列的汉明距离越远，该子序列的丰度就越大。但是，如果受限制的序列是xxy或xyx类型，则会出现非常意外的词频分布。词的丰度很大程度上受相邻词部分重叠限制的影响。例如，当001是限制性限制序列时，距离限制性限制序列两步远的单词100在基因组中变得非常罕见。通过计算可行序列（即没有任何限制位点的序列）中单词的相对丰度，可以很好地解释这种受限制的单词分布。给出了可行序列中字分布的简单公式。

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.