Multiscale Differential Geometry Approaches to Protein Interaction Mechanisms

Epidemic viruses use surface proteins to bind to and invade host cells, which is a critical first step in triggering infection. Understanding how these proteins interact is essential for developing strategies to prevent disease spread and to reveal key biological mechanisms. However, frequent mutations in proteins generate new variants that complicate experimental testing and delay timely responses. This project aims to develop computational mathematical tools to better understand protein structures and interactions, even across a wide range of mutations. The approach combines mathematical modeling, machine learning, and biology to analyze the complex shapes of proteins. These tools will help researchers make faster and more accurate predictions about the infectivity of a given virus strain, potentially enabling quicker public health responses. The project also supports education development by training students in the interdisciplinary studies of mathematics and biology. Overall, this research aims to strengthen society’s ability to anticipate and respond to emerging viral threats by applying efficient and scalable mathematical approaches to pressing biological challenges. Broader impacts include interdisciplinary training and the development of publicly available opensource software to support the biomedical and mathematical sciences communities. This project develops mathematical and computational frameworks for predicting protein-protein binding free energies upon mutations using low-dimensional representations of high-dimensional biomolecular data. Protein-protein interactions play a fundamental role in many biological processes and are particularly critical for mediating viral entry into host cells. However, due to the vast mutation space and the structural complexity of biomolecules, experimental evaluation is limited. The research leverages techniques from algebraic topology, differential geometry, and manifold learning to construct mathematical models that capture both the spatial conformation and physicochemical properties of viral proteins. These models are integrated with deep learning architectures to predict binding affinities efficiently across mutational variants. The novelty lies in the mathematical treatment of biomolecular functions as continuous entities embedded in high-dimensional space, enabling low-dimensional, information-rich manifold representations suitable for learning tasks. The research will develop computational tools that generalize across virus mutations, providing new insight into viral infectivity and epidemiology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.