串联复制的组合数学
The Combinatorics of Tandem Duplication
摘要：
串联复制是一种进化过程，其中一段DNA被复制并近端插入。从这个过程中可能产生的不同配置引起了一些有趣的组合问题。首先，我们引入了一个代数形式主义来表示这个过程作为一个词生产自动机。由n个串联重复产生的单词的数量可以递归地导出。第二，每一个词都有多个演变。借助于双色二维树，构造了对应于一个偏序集的Hasse图，由此可以计算出一个词对应的演化数.第三，我们实现了一些子树修剪和嫁接操作，这个结构上的n个串联重复所产生的可能的进化的总数是$\prod_{k=1}^n(4^k - (2k + 1))$。因此，由串联复制产生的结构的空间以超指数速率增长，其中前导阶项为$\mathcal{O}(4^{\frac{1}{2}n^2})$。
英文摘要：
Tandem duplication is an evolutionary process whereby a segment of DNA is replicated and proximally inserted. The different configurations that can arise from this process give rise to some interesting combinatorial questions. Firstly, we introduce an algebraic formalism to represent this process as a word producing automaton. The number of words arising from n tandem duplications can then be recursively derived. Secondly, each single word accounts for multiple evolutions. With the aid of a bi-coloured 2d- tree, a Hasse diagram corresponding to a partially ordered set is constructed, from which we can count the number of evolutions corresponding to a given word. Thirdly, we implement some subtree prune and graft operations on this structure to show that the total number of possible evolutions arising from n tandem duplications is $\prod_{k=1}^n(4^k - (2k + 1))$. The space of structures arising from tandem duplication thus grows at a super-exponential rate with leading order term $\mathcal{O}(4^{\frac{1}{2}n^2})$.
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