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Much of our mechanistic understanding of the functions of biological macromolecules is based on static structural experiments, which can be modelled either as single structures or conformational ensembles. While these provide us with…
Coarse-graining is a powerful tool for extending the reach of dynamic models of proteins and other biological macromolecules. Topological coarse-graining, in which biomolecules or sets thereof are represented via graph structures, is a…
Unlike conventional two-dimensional (2D) semiconductor superlattices, moir\'{e} patterns in 2D materials are flexible and their electronic, magnetic, optical, and mechanical properties depend on their topography. Within a…
The choice of structural resolution is a fundamental aspect of protein modelling, determining the balance between descriptive power and interpretability. Although atomistic simulations provide maximal detail, much of this information is…
We present a continuum model trained on molecular dynamics (MD) simulations for cellular membranes composed of an arbitrary number of lipid types. The model is constructed within the formalism of dynamic density functional theory and can be…
Membrane proteins deform the surrounding lipid bilayer, which can lead to membrane-mediated interactions between neighboring proteins. Using the mechanosensitive channel of large conductance (MscL) as a model system, we demonstrate how the…
Despite the recognized importance of the multi-scale spatio-temporal organization of proteins, most computational tools can only access a limited spectrum of time and spatial scales, thereby ignoring the effects on protein behavior of the…
Rigidity transitions in simple models of confluent cells have been a powerful organizing principle in understanding the dynamics and mechanics of dense biological tissue. In this work we explore the interplay between geometry and rigidity…
Although RNAs play many cellular functions little is known about the dynamics and thermodynamics of these molecules. In principle, all-atom molecular dynamics simulations can investigate these issues, but with current computer facilities,…
In multi-resolution simulations, different system components are simultaneously modelled at different levels of resolution, these being smoothly coupled together. In the case of enzyme systems, computationally expensive atomistic detail is…
The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes)…
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of…
One of the major shortcomings of discrete element modelling (DEM) is the computational cost required when the number of particles is huge, especially for fine powders and/or industry scale simulations. This study investigates the scaling of…
What does a world model learn from physical exploration, without any linguistic supervision? We argue the answer is organized by a single principle: the geometric structure of the physical world. Training a VAE-based world model on random…
We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of…
Despite recent breakthroughs in understanding how protein sequence relates to structure and function, considerably less attention has been paid to the general features of protein surfaces beyond those regions involved in binding and…
Geometric deep learning has recently achieved great success in non-Euclidean domains, and learning on 3D structures of large biomolecules is emerging as a distinct research area. However, its efficacy is largely constrained due to the…
In our analysis, we show that what Cottenden et al. accomplish is the derivation of the ordinary capstan equation, and a solution to a dynamic membrane with both a zero-Poisson's ratio and a zero-mass density on a rigid right-circular cone.…
Following earlier work with Cris Negron on the cohomology of Drinfeld doubles $D(\mathbb G_{(r)})$, we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" $\tilde D(\mathbb G_{(r)})$ of Frobenius kernels…
Have you ever been troubled by the complexity and computational cost of SE(3) protein structure modeling and been amazed by the simplicity and power of language modeling? Recent work has shown promise in simplifying protein structures as…