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Data-driven discovery and parameter estimation of mathematical models in biological pattern formation [1]

['Hidekazu Hishinuma', 'Department Of Anatomy', 'Cell Biology', 'Graduate School Of Medical Sciences', 'Kyushu University', 'Fukuoka', 'Hisako Takigawa-Imamura', 'Takashi Miura']

Date: 2025-07 

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. 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.

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.

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. Thus, data-driven analysis of mathematical models by comparing target images with pattern images generated by the mathematical models is beneficial in the initial stages of verifying mathematical models. Nevertheless, there has been little research on the methods of such analysis, especially for spatial data. Data-driven analysis of mathematical models involves two issues. One is the selection of mathematical models, identifying candidate models capable of generating patterns similar to the target images. So far, researchers have selected appropriate mathematical models empirically. This process should be more automated since different mechanisms can lead to similar patterns [1] and there are a wide variety of mathematical models to consider. Currently, selecting mathematical models is a less challenging problem than formulating the models themselves. It has been difficult to formulate mathematical models for target images flexibly using a machine learning approach. In a previous study of formulating ordinary differential equations by a machine learning approach, a large amount of time-series data is required to predict mathematical models [2]. However, only steady-state images are often available without time-series data in experiments. There are a few methods based only on steady-state images. For example, in material science, a method to formulate the appropriate model from target images was developed using the knowledge of crystallographic symmetry [3], whereas such knowledge is often not available in biology. The second issue is parameter estimation for the mathematical model of interest. Even with the same model parameters, the model may generate slightly different images if it relies on randomly set initial values. Therefore, it is desirable to predict parameters from a few steady-state images that differ slightly depending on the initial conditions. There are several studies that can deal with parameter estimation and were tested with Turing models [1, 4]. When estimating the parameters, quantifying the uncertainty of the prediction is also useful for gaining insights into the parameters since it enables us to evaluate the robustness to perturbations of each parameter and the validity of the mathematical model. Some of the previous studies above have carried out this with the Bayesian approach [1, 3]. We propose a less constrained method that enables parameter prediction in a short time from only a few target images by completely decoupling the training and prediction stages in the approximate Bayesian estimation algorithm. Here, we propose a comprehensive method to accomplish both the selection and parameter estimation of mathematical models for two-dimensional patterns by using machine learning. To address the problems, we embed the target or generated pattern images into vectors, which reflect the essential features of the images, by a deep learning model. The vectors enable the selection of various mathematical models and Bayesian parameter estimation considering uncertainty of the prediction by using only a few steady-state images.

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. We enabled feature extraction by mapping patterns from various mathematical models to the same latent space by utilizing the foundation model ViT without fine-tuning. We demonstrated that the embedding by the neural network appropriately functions as feature extraction for images. Also, we developed an approximate Bayesian estimation of parameters by interpreting the regression model NGBoost, which outputs prediction and uncertainty, within the context of Bayesian statistics. Mathematical models can generate synthetic data depending on the model parameters, but the parameters usually cannot be predicted explicitly from the data because their likelihood is typically intractable. Such scenarios have led to the development of Bayesian inference approaches known as likelihood-free inference or Simulation-Based Inference, which have been actively researched in recent years [21]. Previous studies have not completely separated the data generation process from the inference process [22]. Our method requires only a single training session on a small dataset and does not necessitate the additional generation of pattern images during the inference phase. This efficiency is particularly advantageous when iterative inferences are required for a single mathematical model. In the current implementation, we opted for NGBoost because the machine learning model must account for the uncertainty of predictions. Although NGBoost is a lightweight and excellent model, we have to determine the output format as a single parametric probability distribution. Therefore, if the predetermined probability distribution significantly differs from the true distribution, prediction accuracy may decrease. This is suggested by the results in the Supporting information(S6 Fig). To resolve this issue, the predictive model for parameters needs to be able to learn any probability distribution such as normalization flow [23]. This method can be applied to various pattern formation phenomena in life science. For example, although the Gray-Scott model is a famous pattern formation model, a good biological application has yet to be found. Our method will help discover an appropriate biological system to which we can apply this model. In addition, parameter estimation enables us to validate the models by comparing parameter space size and pattern frequency and by predicting pattern change following experimental perturbations. Confirming the effectiveness of this method through applications to other biological patterns is an important challenge for the future. Given the constraints of data obtained from biological observations, we proposed the methods for screening candidate mathematical models and estimating specific parameters using only a limited number of snapshots. However, when multiple mathematical models can theoretically generate identical steady-state patterns—such as the Model B Cahn-Hilliard equation and the Model H advective Cahn-Hilliard equation, it may not be feasible to differentiate between them using steady-state data alone. In these cases, it becomes necessary to incorporate dynamic data, such as time-series information, for accurate model identification. Several promising approaches, including Physics-Informed Neural Network(PINN) [24] and Sparse Identification of Nonlinear Dynamics (SINDy) [2], have been developed to generate governing equations that are well-suited for scenarios where time-series data is available. These methods present viable options for dynamic data. Nonetheless, our parameter estimation approach also has the potential to be extended to such data. By applying NGBoost, implemented within SD-NPE, to data at arbitrary time points, it may be possible to resolve this challenge. However, SD-NPE is based on the assumption of independence between data points. Since time-series data inherently involves dependencies, the formation of a latent space that accounts for temporal information would be essential for accurate Bayesian estimation. Future work could focus on adapting SD-NPE to accommodate these dependencies, enabling it to effectively process dynamic data in a temporally-aware latent space. The emphasis on both the deductive approach of mathematical models and the inductive approach of machine learning could compensate for each other’s weaknesses. This study offers opportunities for data-driven preliminary screening or validation of mathematical models. Mathematical models are well explainable but require theoretical understanding and domain-specific knowledge for their design. On the other hand, machine learning can provide accurate predictions that fit the data it has learned, though those are not explainable. Therefore, we thought the integration of these two methods was a significant issue in elucidating mechanisms that can accurately explain observational data in the field of mathematical biology. So far, there have been few such studies [25]. Through this research, we have presented a new direction in the analysis of mathematical models using machine learning.

Limitations of the study Preprocessing can significantly affect feature extraction into embedding vectors by CLIP. Without proper preprocessing, incorrect features can be caught by CLIP, and that may adversely affect model selection and parameter estimation. Currently, it is challenging to robustly extract information about abstract geometric patterns of interest from images in nature, and improvements are anticipated in future research. Our method only applies to two-dimensional data, as the feature extraction process relies on the ViT model. To extend the method to three-dimensional data or other spatially structured data, such as spherical surfaces, ViT must be replaced with an appropriate encoder that supports these data formats. Furthermore, there is a practical limit to the number of parameters that can be predicted due to computational constraints. We examined the relationship between the number of unknown parameters and the required training dataset size through parameter estimation in linear regression models (S2 Fig). The results suggest that the impact of the number of parameters on dataset size requirements is not substantial. However, as mentioned in the original NGBoost paper [7], the computational cost of the algorithm scales cubically with the number of parameters and linearly with the dataset size. This implies that executing the model becomes challenging when the number of parameters exceeds five. To address this limitation, an alternative estimator with a lower computational cost should be adopted.

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.

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