A Prüfer-Sequence Based Algorithm for Calculating the Size of Ideal Randomly Branched Polymers.

Area of Science:

• Polymer Science
• Graph Theory
• Computational Chemistry

Background:

• Branched polymers can be modeled as tree graphs.
• Prüfer sequences provide a unique representation for labeled tree graphs.
• Permutations of Prüfer sequences generate trees with identical degree distributions but varying structures.

Purpose of the Study:

• To develop a method for generating random branched polymers with controlled degree distributions.
• To efficiently calculate graph distances directly from Prüfer sequences.
• To investigate the 3D size scaling laws of branched polymers using this novel approach.

Main Methods:

• Representing branched polymers as labeled tree graphs.
• Utilizing Prüfer sequences and their permutations to generate ensembles of random tree graphs.
• Developing an algorithm to compute graph distances from Prüfer sequences.
• Calculating 3D polymer size metrics (radius of gyration, average end-to-end distance) from graph distances.

Main Results:

• Generated large ensembles of random tree graphs with identical degree distributions.
• Demonstrated efficient calculation of graph distances directly from Prüfer sequences.
• Showed that 3D size measures for ideal randomly branched polymers follow an N(1/4) scaling law.
• Observed that random-sequence RNA molecules exhibit N(1/3) scaling for radius of gyration before structural randomization.

Conclusions:

• The Prüfer sequence method provides a powerful tool for studying branched polymer structures and properties.
• The N(1/4) scaling law appears to be a universal characteristic for randomly branched polymers.
• This approach offers new insights into the relationship between sequence, structure, and physical properties of polymers, including biomolecules like RNA.