前生物进化的置换不变Crow-Kimura模型的线性代数
Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution
摘要：
著名的准种模型的一个特殊情况下-的Crow-Kimura模型与置换不变的适应度景观-进行了研究。利用置换不变的适应度景观的情况下的突变矩阵具有特殊的三对角形式的事实，建议的基础的变化，使得在新的坐标可以得到一些分析结果。特别是，使用的突变矩阵的特征向量作为新的基础上，我们表明，准种分布接近二项式，并给出简单的估计的收敛速度。建议的方法的另一个后果是确定准物种的方程组的参数解。使用这个参数的解决方案，我们表明，我们的方法导致精确的渐近结果在某些情况下，这是不包括现有的方法。特别是，我们不仅能够提出的领导特征值（平均人口适应度）的极限行为，但也极限quasispecies特征向量的特殊情况下的精确公式。例如，在经典单峰适应度景观的情况下，该特征向量具有几何分布。在生物学方面，我们提出了一个数学定义，准物种的二项分布，它可以被用作臭名昭著的错误阈值的操作定义的接近的基础上。利用这个定义，我们提出了两个近似公式来估计临界突变率后，准种离域发生。
英文摘要：
A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.
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