Stability and adaptability are two conflicting requirements of living systems. Previous statistical survey on the structural ensembles of folded proteins showed that natural proteins have evolved to critical point, at which the proteins have the maximal adaptability but simultaneously maintaining the structural integrity. Here we study the correlation and fluctuation properties of natural proteins from the sequence level by using complex network methods. By performing direct coupling analysis to the amino acid sequences of homologous proteins, we constructed the coevolution networks for the protein families and analyzed the statistical and correlation properties. The results showed that the edge weights of the networks, which were characterized by the direct information of the coevolutionary analysis, have power law distributions. In addition, the correlation length of the fluctuations is proportional to the topological sizes of the proteins, demonstrating scale-free feature of the correlated fluctuations. Our results provide new signature of critical behaviors of the proteins based on the information of amino acid sequences, supporting the previous proposal that natural proteins have evolved to the vicinity of critical point.

稳定性和适应性是生命系统的两个相互矛盾的要求。先前对折叠蛋白的结构整体进行的统计调查表明，天然蛋白已进化到临界点，在该临界点处，蛋白具有最大的适应性，但同时又保持了结构完整性。在这里，我们使用复杂的网络方法从序列水平研究天然蛋白的相关性和起伏特性。通过对同源蛋白质的氨基酸序列进行直接偶联分析，我们构建了蛋白质家族的协同进化网络，并分析了统计和相关特性。结果表明，以协进化分析的直接信息为特征的网络边缘权重具有幂律分布。此外，波动的相关长度与蛋白质的拓扑大小成正比，证明了相关波动的无标度特征。我们的结果基于氨基酸序列的信息为蛋白质的关键行为提供了新的特征，支持了先前的建议，即天然蛋白质已经进化到临界点附近。