Background and Challenges

Recent breakthroughs in measurement technologies in the life and biomedical sciences have generated an unprecedented diversity of data, revealing the intrinsic complexity and dynamic nature of various biological phenomena. As a mathematical foundation for leveraging such data, geometric analysis has found active applications in analyzing single-cell gene expression data. This is partly owing to its ability to handle distances between probability distributions and their dynamic transformations from the perspectives of optimal transport and gradient flows.

However, the data produced by emerging technologies are not limited to continuous ones like gene expression. Discrete biological information—such as symbolic data from DNA sequences and discrete chromatin states—also plays a critical role. Traditional geometric analysis, which assumes an underlying continuous manifold, is not naturally suited to such data. To apply it, discrete data must first be encoded into a continuous representation, and then the analytical results must be decoded back into discrete form. This intermediate step means that the underlying structure and informational content inherent to the discrete data are not directly and fully utilized. Moreover, the analysis of discrete data, such as DNA sequences, has developed largely outside the geometric analysis framework rooted in bioinformatics, resulting in disciplinary separation. Thus, the fundamental challenge is to establish an integrative framework for the geometric analysis of both discrete and continuous data.

Beyond just handling data, biological phenomena such as intracellular reactions present additional theoretical challenges that traditional geometric analysis cannot capture sufficiently. Relationships among biological entities—molecules, cells, organs—often form nonlinear and many-to-many causal structures represented by hypergraphs. Yet, mathematical methodologies to analyze such topological and geometric properties remain underdeveloped. Furthermore, the dynamics typically exploited in geometric analysis are symmetric and amenable to gradient flow formulations. In contrast, complex biological dynamics often exhibit inherent asymmetries and irreversibilities originating from violation of detailed balance, which give rise to oscillatory or chaotic behaviors that deviate from gradient-flow paradigms. How to handle such dynamics from the viewpoint of geometric analysis, and how to connect such treatments to data analysis and biological understanding, remains a major open challenge.

Objective

In light of the aforementioned background, the prediction and control of complex biological dynamics and their transformations and breakdowns require a geometric perspective and mathematical foundation capable of encompassing both continuous and discrete structures, as well as capturing many-to-many, nonlinear, and asymmetric relationships. In this CREST project, we aim to establish an interdisciplinary research framework that integrates pure mathematics, applied mathematics, and informatics to pursue the following goals:

[Mission 1] Develop a new geometric analysis theory that can address discreteness, nonlinearity, and asymmetry, which are essential features of biological systems;

[Mission 2] Utilize the developed theoretical framework to build information technologies that predict real-world phenomena such as the collapse of inter-organ interactions, the breakdown of immune memory and homeostasis, and the transitions in cellular differentiation states;

[Mission 3] Foster a new research paradigm and community that integrates "discrete and continuous" geometric analysis, and also that bridges "fundamental theory and practical application," through collaborations between mathematicians, data scientists, and biomedical researchers. 

[Mission 1] Development of a Geometric Analysis Framework for Discreteness, Nonlinearity, and Asymmetry

To address geometric properties unique to discrete structures and inherent asymmetries, it is essential to advance theoretical foundations that can capture topological features of hypergraphs representing many-to-many relations, extend concepts of dual flat structures manifesting on discrete spaces, and deepen the mathematical connections between discrete and continuous geometric analysis.

This project will integrate several recent advances: Ohta's work on nonlinear Laplacians on hypergraphs and Ricci curvature theories based on optimal transport; generalizations of dual flat structures via Kähler and contact geometry; and insights from hypergraph topologies in reaction networks. Together, these components will contribute to the advancement of discrete geometric analysis.

A particular focus will be placed on the treatment of asymmetry, a challenge shared by both discrete and continuous settings. Ohta's group will approach the problem through extending gradient flow theories to asymmetric metric spaces, including Finsler manifolds. In parallel, Kobayashi's group will investigate mathematical frameworks that incorporate nonequilibrium flows induced by asymmetry, drawing on concepts such as metriplectic structures and Laplacians with magnetic fields. By elucidating the interplay between these approaches, we aim to construct a novel geometric theory capable of handling asymmetries and the complex dynamics they entail.

[Mission 2] Development of Integrated Data Analysis Methods for Prediction and Control of Complex Biological Dynamics and Disease Progression

The states of cells and tissues are characterized not only by continuous data such as gene expression, but also by discrete structural data—including gene regulatory interactions, chromatin configurations, and repertoires of DNA or immune receptor sequences. In addition, the latent asymmetries in such discrete data often underlie system instability and dynamic transitions.

Building on single-cell RNA-seq data analysis by Honda's group and immune receptor and sequence-based data analysis by Kobayashi's group, we will develop integrative methods that combine continuous and discrete information to characterize biological systems and predict or control their transformations and breakdowns.

Specifically, we will target phenomena such as inter-organ communication, immune homeostasis, and cell differentiation processes driven by asymmetric dynamic attractors. To this end, we will develop new methods in discrete-continuous optimal transport, system estimation for discrete-continuous hybrid systems, and large deviation theory and optimal control for discrete-continuous stochastic processes. The validity of these methods will be tested on real biological datasets, ultimately leading to a robust methodology for inferring system states and transitions from data.

[Mission 3] Establishment of Research Community Integrating “Discrete and Continuous” and “Theory and Application” in Geometric Analysis

Through geometric analysis as a unifying framework, we will build a collaborative system connecting researchers from mathematics to biomedical sciences. Based on the theoretical developments in Mission 1, we will deepen the mathematical foundations of the integrative geometric analysis that handles both discrete and continuous aspects.

In parallel, by leveraging the applied developments in Mission 2 and actively interacting with experimental and biomedical research communities, we will promote translational applications of our methods. By integrating these efforts, we aim to foster a new scientific paradigm and research community that spans both "discrete and continuous" and "fundamental and applied" domains in geometric analysis.

Future Outlook

Geometric analysis on continuous spaces has historically developed in conjunction with physics and engineering, and is now being actively applied in other fields such as information engineering and data science. In contrast, fundamentally discrete natures are manifested in many phenomena and datasets from life sciences, medicine, and human society—such as inter-organ, individual, or societal networks, as well as chemical molecular structures.

Thus, the theoretical and technological foundations established in this CREST project are not limited to applications in biomedical sciences. They have the potential to be applied to a wide range of life and social science domains, including ecosystem collapse, the prediction and control of epidemic dynamics, and crowd behavior, as well as chemo-informatics.

Moreover, the interdisciplinary team structure—spanning pure mathematics to medical science—will foster the development of human resources and methodological expertise capable of addressing such integrated challenges. This framework is expected to serve as a springboard for future research initiatives that go beyond the scope of the present project.