Tim Marshall | Search arXiv

arXiv2026

Random complex cross ratios

We study the cross ratio of four independent points chosen uniformly on the Riemann sphere. A positive integral over a two-dimensional simplex gives its normalized planar density. We then evaluate this integral in terms of the Bloch--Wigner dilogarithm, providing a closed formula. We describe its symmetries, real-axis values, moment bounds and geometric interpretations. We use these to evaluate the mean ideal-tetrahedron volume directly from the density, recovering $π/6$, a recent theorem of Kabluchko and Schange. We then prove that the expected distance between two independently sampled hyperbolic geodesics in $\mathbb{H}^3$ is $4/3$, by evaluating the logarithmic potential of the cross-ratio distribution on the negative real axis. Further applications give the joint dihedral-angle density and an exact acute-tetrahedron probability, and a branched-cover probability law on the modular surface with exponentially small cusp tails.

math.CV↗

arXiv2025

Automatic Forward Model Parameterization with Bayesian Inference of Conformational Populations

To quantify how well theoretical predictions of structural ensembles agree with experimental measurements, we depend on the accuracy of forward models. These models are computational frameworks that generate observable quantities from molecular configurations based on empirical relationships linking specific molecular properties to experimental measurements. Bayesian Inference of Conformational Populations (BICePs) is a reweighting algorithm that reconciles simulated ensembles with ensemble-averaged experimental observations, even when such observations are sparse and/or noisy. This is achieved by sampling the posterior distribution of conformational populations under experimental restraints as well as sampling the posterior distribution of uncertainties due to random and systematic error. In this study, we enhance the algorithm for the refinement of empirical forward model (FM) parameters. We introduce and evaluate two novel methods for optimizing FM parameters. The first method treats FM parameters as nuisance parameters, integrating over them in the full posterior distribution. The second method employs variational minimization of a quantity called the BICePs score that reports the free energy of `turning on` the experimental restraints. This technique, coupled with improved likelihood functions for handling experimental outliers, facilitates force field validation and optimization, as illustrated in recent studies (Raddi et al. 2023, 2024). Using this approach, we refine parameters that modulate the Karplus relation, crucial for accurate predictions of J-coupling constants based on dihedral angles between interacting nuclei. We validate this approach first with a toy model system, and then for human ubiquitin, predicting six sets of Karplus parameters. Finally, we demonstrate that our framework naturally generalizes optimization to any differentiable forward model...

physics.bio-ph↗