Singaram Surendra W, Gopal Ajaykumar, Ben-Shaul Avinoam
Institute of Chemistry and the Fritz Haber Research Center, Givat Ram Safra Campus, The Hebrew University , Jerusalem 91904, Israel.
Department of Chemistry, University of California , Los Angeles, California 90095, United States.
J Phys Chem B. 2016 Jul 7;120(26):6231-7. doi: 10.1021/acs.jpcb.6b02258. Epub 2016 May 10.
Branched polymers can be represented as tree graphs. A one-to-one correspondence exists between a tree graph comprised of N labeled vertices and a sequence of N - 2 integers, known as the Prüfer sequence. Permutations of this sequence yield sequences corresponding to tree graphs with the same vertex-degree distribution but (generally) different branching patterns. Repeatedly shuffling the Prüfer sequence we have generated large ensembles of random tree graphs, all with the same degree distributions. We also present and apply an efficient algorithm to determine graph distances directly from their Prüfer sequences. From the (Prüfer sequence derived) graph distances, 3D size metrics, e.g., the polymer's radius of gyration, Rg, and average end-to-end distance, were then calculated using several different theoretical approaches. Applying our method to ideal randomly branched polymers of different vertex-degree distributions, all their 3D size measures are found to obey the usual N(1/4) scaling law. Among the branched polymers analyzed are RNA molecules comprised of equal proportions of the four-randomly distributed-nucleotides. Prior to Prüfer shuffling, the vertices of their representative tree graphs, these "random-sequence" RNAs exhibit an Rg ∼ N(1/3) scaling.
支化聚合物可以表示为树形图。由N个标记顶点组成的树形图与N - 2个整数序列（称为普吕弗序列）之间存在一一对应关系。该序列的排列产生对应于具有相同顶点度分布但（通常）不同分支模式的树形图的序列。通过反复洗牌我们生成的普吕弗序列，得到了大量具有相同度分布的随机树形图集合。我们还提出并应用了一种有效的算法，直接从普吕弗序列确定图距离。然后，使用几种不同的理论方法，从（由普吕弗序列导出的）图距离计算3D尺寸度量，例如聚合物的回转半径Rg和平均端到端距离。将我们的方法应用于具有不同顶点度分布的理想随机支化聚合物，发现它们所有的3D尺寸度量都遵循通常的N(1/4)标度律。在分析的支化聚合物中，有由四种随机分布核苷酸等比例组成的RNA分子。在普吕弗洗牌之前，这些“随机序列”RNA的代表性树形图的顶点呈现出Rg ∼ N(1/3)标度。
