Data-driven modeling and deep learning present powerful tools that are opening up new paradigms and opportunities in the understanding, discovery, and design of soft and biological materials. I will describe our recent applications of deep representational learning to expose the sequence-function relationship within homologous protein families and to use these principles for the data-driven design and experimental testing of synthetic proteins with elevated function. I will then describe an approach based on latent space simulators to learn ultra-fast surrogate models of protein folding and biomolecular assembly by stacking three specialized deep learning networks to (i) encode a molecular system into a slow latent space, (ii) propagate dynamics in this latent space, and (iii) generatively decode a synthetic molecular trajectory.

Abstract: Neurotransmission at chemical synapses relies on the calcium-induced fusion of synaptic vesicles with the presynaptic membrane. The distance of the vesicle to the calcium channels determines the fusion probability and consequently the postsynaptic signal. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. For all these process components, stochastic effects are widely recognized to play an important role. In this talk, I will present our recent efforts on how to describe and structurally understand neurotransmission dynamics using stochastic modeling approaches. Starting with a linear reaction scheme, a method to directly compute the exact first- and second-order moments of the filtered output signal is proposed. For a modification of the model including explicit recovery steps, the stochastic dynamics are compared to the mean-field approximation in terms of reaction rate equations. Finally, we reflect on spatial extensions of the model, as well as on their approximation by hybrid methods.

References:

- A. Ernst, C. Schütte, S. Sigrist, S. Winkelmann. Mathematical Biosciences, 343, 108760, 2022.

- A. Ernst, N. Unger, C. Schütte, A. Walter, S. Winkelmann. Under Review. https://arxiv.org/abs/2302.01635

There has been a great deal of interest in understanding the link between the axiomatic descriptions of conformal field theory given by vertex operator algebras and conformal nets. In recent work, we establish an equivalence between certain vertex algebras and conformally-symmetric quantum field theories in the sense of Wightman. In this talk I will give an overview of these results and discuss some of the difficulties that arise, the functional analytic properties of vertex algebras, and some of the ideas for future work in this area.

This is joint work with James Tener and Yoh Tanimoto.

The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

We propose a nematic model for polyatomic gas, intending to study anisotropic phenomena. Such phenomena stem from the orientational degree of freedom associated with the rod-like molecules composing the gas. We adopt as a primer the Curitss-Boltzmann equation. The main difference with respect to Curtiss theory of hard convex body fluids is the fact that the model here presented takes into account the emergence of a nematic ordering. We will also derive from a kinetic point of view an energy functional similar to the Oseen-Frank energy. The application of the Noll-Coleman procedure to derive new expressions for the stress tensor and the couple-stress tensor will lead to a model capable of taking into account anisotropic effects caused by the emergence of a nematic ordering. In the near future, we hope to adopt finite-element discretisations together with multi-scale methods to simulate the integro-differential equation arising from our model.

This talk presents a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and the market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the  order book evolution into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics. This talk is based on joint work with Rama Cont and Pierre Degond.