Data-driven Discovery and Parameter Estimation of Mathematical Models in Biological Pattern Formation
Hidekazu Hishinuma, Hisako Takigawa-Imamura, Takashi Miura.
Abstract
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.
Introduction
A wide variety of spatial patterns are found in living organisms. Various mathematical models have been proposed to gain insight into the mechanism of these pattern formations. These mathematical models are constructed based on knowledge and hypotheses related to the morphogenesis. It is important to verify that the mathematical models can appropriately explain the morphogenesis of interest, whereas experimental verification is difficult and incurs high costs.
Methods:
In the selection of mathematical models, the first step is feature extraction. A target image was encoded into a 512-dimensional vector by the Vision Transformer (ViT) [5] image encoder, extracted from the original Contrastive Language-Image Pre-training (CLIP) model [6] and used in a zero-shot setting, meaning it was applied without any fine-tuning. Then, the similarities between the embedding vector of the target image and those of dataset including pattern images of various mathematical models were calculated. Finally, the pattern images of mathematical models with high similarity were selected.
Discussion:
In this study, we developed general methods for identifying mathematical models that exhibit patterns of interest, and for performing Bayesian estimation of the parameters of a given mathematical model. In the machine learning field, the general feature extraction method from pattern images and the cost optimization of approximate Bayesian estimation are two major challenges.
Acknowledgments:
We are grateful to MARINE WORLD uminonakamichi aquarium for granting us the opportunity to photograph fish, which enhanced the quality of this research. We also thank them for their permission to use these images in our publication.
Citation: Hishinuma H, Takigawa-Imamura H, Miura T (2025) Data-driven discovery and parameter estimation of mathematical models in biological pattern formation. PLoS Comput Biol 21(1): e1012689. https://doi.org/10.1371/journal.pcbi.1012689
Editor: Miguel Francisco de Almeida Pereira de Rocha, University of Minho, PORTUGAL
Received: May 14, 2024; Accepted: December 2, 2024; Published: January 23, 2025.
Copyright: © 2025 Hishinuma et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The source code used in this study are available on GitHub at https://github.com/Hide-Hishi/CLIP-MMA. The dataset and trained models are available on FigShare at https://doi.org/10.6084/m9.figshare.27263601.v1 and https://doi.org/10.6084/m9.figshare.27263634.v1.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
