任意大小字母表的Crow-Kimura和本征模型的Schwinger自旋相干态解
Solution of the Crow-Kimura and Eigen models for alphabets of arbitrary size by Schwinger spin coherent states
摘要：
为了表示核酸和蛋白质序列的进化，我们表示的平行和本征模型的分子进化的功能积分表示与$h$字母表，解除两个国家，嘌呤/嘧啶的假设，往往在准物种理论。对于任意的$h$和一般的变异方案，我们得到了该模型的最大值原理的解决方案。欧拉的齐次函数定理是用来推导这种“热力学”的演变公式。平行模型的一般结果简化为Kimura 3 ST突变方案的嘌呤/嘧啶$h=2$字母表和核酸$h=4$字母表的已知结果。举例说明$h=4$和$h=20$的情况。我们导出了一般$h$的本征模型的最大值原理。本征模型的一般结果简化为$h=2$的已知结果。给出了核酸$h=4$和氨基酸$h=20$字母表的实例。在这些模型中发生了错误突变相变，并且当字母表大小增加到超过两个字母时，相变的顺序从二阶变为一阶。作为例子，我们分析了一般的解析解尖峰，线性，二次和四次适应度函数。
英文摘要：
To represent the evolution of nucleic acid and protein sequence, we express the parallel and Eigen models for molecular evolution in terms of a functional integral representation with an $h$-letter alphabet, lifting the two-state, purine/pyrimidine assumption often made in quasi-species theory. For arbitrary $h$ and a general mutation scheme, we obtain the solution of this model in terms of a maximum principle. Euler's theorem for homogeneous functions is used to derive this `thermodynamic' formulation of evolution. The general result for the parallel model reduces to known results for the purine/pyrimidine $h=2$ alphabet and the nucleic acid $h=4$ alphabet for the Kimura 3 ST mutation scheme. Examples are presented for the $h=4$ and $h=20$ cases. We derive the maximum principle for the Eigen model for general $h$. The general result for the Eigen model reduces to a known result for $h=2$. Examples are presented for the nucleic acid $h=4$ and the amino acid $h=20$ alphabet. An error catastrophe phase transition occurs in these models, and the order of the phase transition changes from second to first order for smooth fitness functions when the alphabet size is increased beyond two letters to the generic case. As examples, we analyze the general analytic solution for sharp peak, linear, quadratic, and quartic fitness functions.
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