Comparing and distinguishing the structure of biological branching

Bifurcating developmental branching morphogenesis gives rise to complex organs such as the lung and the ureteric tree of the kidney. However, a few quantitative methods or tools exist to compare and distinguish, at a structural level, the critical features of these important biological systems. Here we develop novel graph alignment techniques to quantify the structural differences of rooted bifurcating trees and demonstrate their application in the analysis of developing kidneys from in normal and mutant mice. We have developed two graph based metrics: graph discordance, which measures how well the graphs representing the branching structures of distinct trees graphs can be aligned or overlayed; and graph inclusion, which measures the degree of containment of a tree graph within another. To demonstrate the application of these approaches we first benchmark the discordance metric on a data set of 32 normal and 28Tgfβ+/− mutant mouse ureteric trees. We find that the discordance metric better distinguishes control and mutant mouse kidneys than alternative metrics based on graph size and fingerprints - the distribution of tip depths. Using this metric we then show that the structure of the mutant trees follows the same pattern as the normal kidneys, but undergo a major delay in elaboration at later stages. Analysis of both controls and mutants using the inclusion metric gives strong support to the hypothesis that ureteric tree growth is stereotypic. Additionally, we present a new generalised multi-tree alignment algorithm that minimises the sum of pairwise graph discordance and which can be used to generate maximum consensus trees that represent the archetype for fixed developmental stages. These tools represent an advance in the analysis and quantification of branching patterns and will be invaluable in gaining a deeper understanding of the mechanisms that drive development. All code is being made available with documentation and example data with this publication.