Research Program & Open Problems

10.1 Open Problems

Problem 1 — Exact MAML–NZ correspondence beyond the Gaussian approximation

The formal identification of the MAML inner loop with the NZ memory integral was established under the assumption of Gaussian task distributions and quadratic loss landscapes. For non-Gaussian task ensembles (e.g., protein families with rare but functionally important conformations), the correspondence requires corrections from higher cumulants of the gradient distribution. Deriving these corrections and testing them numerically on protein benchmark datasets is an open problem of both mathematical and practical significance.

Problem 2 — Beyond the cusp: swallowtail and butterfly catastrophes

The cusp catastrophe (codimension 2) is the appropriate universality class for proteins with one fast and one slow conformational coordinate. Allosteric proteins with multiple slow modes may require higher-order catastrophes: the swallowtail (codimension 3) or butterfly (codimension 4). Does the MAML loss landscape exhibit higher-order catastrophes for multi-domain proteins? This requires extending the Hessian spectroscopy of Module 6 to monitor multiple eigenvalues simultaneously, and connecting the observed bifurcation structure to the allosteric communication network.

Problem 3 — MAML for Schrödinger-bridge path sampling

The Schrödinger bridge (SB) problem asks for the most probable stochastic process connecting two endpoint distributions — directly relevant to protein folding and conformational transitions. Recent work connects SB to score-based diffusion models. Can MAML provide a meta-initialisation for the SB drift function that adapts to a new protein transition with few trajectory samples? This would combine the memory-kernel interpretation of Module 5 with the path-sampling power of SB methods, potentially enabling rapid characterisation of rare conformational transitions.

Problem 4 — Quantum memory kernels: NMQJ vs. NZ in enzymatic tunneling

The NZ equation treats the bath quantum mechanically but the system–bath coupling perturbatively. For enzymatic proton tunneling, the coupling between the transferring proton and protein scaffold modes can be non-perturbative (strong-coupling limit, polaron physics). The Non-Markovian Quantum Jump (NMQJ) method offers an alternative trajectory-based unraveling that is non-perturbative. Does the MAML–NMQJ correspondence still hold? What is the physical content of the meta-initialisation in the strong-coupling limit?

Problem 5 — Experimental validation via site-directed mutagenesis

The tunneling-suppression hypothesis predicts that mutations which increase scaffold rigidity (e.g., proline substitutions, disulfide crosslinks, or designed β-sheet reinforcement) should reduce KIE toward the classical limit. Conversely, flexibility-enhancing mutations (Gly substitutions, helix-breaking mutations) should increase KIE. These are testable predictions — but require collaboration between computational and experimental groups. The MAML framework makes quantitative predictions (Eq. 9.1) that could be directly tested via stopped-flow kinetics and primary KIE measurements on engineered enzyme variants.

10.2 Connecting to the JCP “Memory in Biomolecular Dynamics” Special Topic

The theoretical framework developed in this course constitutes the conceptual foundation for submission to the Journal of Chemical Physics special topic “Memory in Biomolecular Dynamics” (August 2026 deadline). The manuscript targets three contributions:

1. Formal: The NZ–MAML correspondence (Module 5).
2. Computational: MAML-NequIP RPMD KIE predictions for the M0 → IFP benchmark suite (Module 7).
3. Interpretive: The cusp catastrophe as a unified framework for bifurcations in both the physical energy landscape and the MAML loss landscape (Module 6).