Data-driven discovery and parameter estimation of mathematical models in biological pattern formation

Mathematical modeling has been utilized to explain biological pattern formation, but the selections of models and parameters have been made empirically. In the present study, we propose a data-driven approach to validate the applicability of mathematical models. Specifically, we developed methods to automatically select the appropriate mathematical models based on the patterns of interest and to estimate the model parameters. For model selection, we employed Contrastive Language-Image Pre-training (CLIP) for zero-shot feature extraction, mapping the given pattern images to latent space and specifying the appropriate model. For parameter estimation, we developed a novel technique that rapidly performs approximate Bayesian inference based on Natural Gradient Boosting (NGBoost). This method allows for parameter estimation under minimal constraints; i.e., it does not require time-series data or initial conditions and is applicable to various types of mathematical models. We tested the method with Turing patterns and demonstrated its high accuracy and correspondence to analytical features. Our strategy enables efficient validation of mathematical models using spatial patterns.Author summary: Biological systems show various beautiful patterns, and diverse mathematical models have been proposed to gain deep insights into the mechanisms behind the pattern formation. For example, animal coat markings show variety of attractive patterns that can be generated by Turing model. However, selecting the candidate models has been done empirically, and the experimental estimation of parameters has been costly. Recently, machine learning technologies have made remarkable progress, solving various tasks of image recognition. In this study, we utilize this machine learning technology and propose two novel data-driven methods: a method for selecting mathematical models that can generate observed patterns, and a method for estimating the parameters of the mathematical model. Using a foundation model, we convert observed patterns and mathematical model patterns into vectors in a common latent space, and the candidate mathematical models are selected according to the similarity between these vectors. Parameter estimation is performed by dimensionally reducing the vectors and inputting them into approximate Bayesian inference. We validate our method with Turing patterns, confirming that this method aligns with human visual perception and that model parameters could be estimated with high accuracy.