Quantifying Different Modeling Frameworks Using Topological Data Analysis: A Case Study with Zebrafish Patterns

Mathematical models come in many forms across biological applications. In the case of complex, spatial dynamics and pattern formation, stochastic models also face two main challenges: pattern data are largely qualitative, and model realizations may vary significantly. Together these issues make it difficult to relate models and empirical data---or even models and models---limiting how different approaches can be combined to offer new insights into biology. These challenges also raise mathematical questions about how models are related, since alternative approaches to the same problem---e.g., Cellular Potts models; off-lattice, agent-based models; on-lattice, cellular automaton models; and continuum approaches---treat uncertainty and implement cell behavior in different ways. To help open the door to future work on questions like these, here we adapt methods from topological data analysis and computational geometry to quantitatively relate two different models of the same biological process in a fair, comparable way. To center our work and illustrate concrete challenges, we focus on the example of zebrafish-skin pattern formation, and we relate patterns that arise from agent-based and cellular automaton models.