Normalized Topological Indices Discriminate between Architectures of Branched Macromolecules

Branching architecture characterizes numerous systems, ranging from synthetic (hyper)branched polymers and biomolecules such as lignin, amylopectin, and nucleic acids to tracheal and neuronal networks. Its ubiquity reflects the many favorable properties that arise because of it. For instance, branched macromolecules are spatially compact and have a high surface functionality, which impacts their phase characteristics and self-assembly behavior, among others. The relationship between branching and physical properties has been studied by mapping macromolecules to mathematical trees whose architecture can be characterized using topological indices. These indices, however, do not allow for a comparison of macromolecules that map to trees of different sizes, be it due to different mapping procedures or differences in their molecular weight. To alleviate this, we introduce a novel normalization of topological indices using estimates of their probability density functions. We construct a phase space using two normalized topological indices, which enables a robust discrimination between different architectures of branched macromolecules. We demonstrate the necessity of such a phase space on two practical applications, one being ribonucleic acid molecules with various branching topologies and the other different methods of coarse-graining branched macromolecules. Our approach can be applied to any type of branched molecule and extended as needed to other topological indices, making it useful across a wide range of fields where branched molecules play an important role, including polymer physics, green chemistry, bioengineering, biotechnology, and medicine.