A Dual-Boundary-Tracking Finite Element Method for the Lamm Equation in Sedimentation Velocity Analytical Ultracentrifugation
Abstract
Analytical ultracentrifugation (AUC) is widely used to characterize the size distributions of proteins and other biological macromolecules. Quantitative analysis of sedimentation velocity data requires numerical solution of the finite-column Lamm equation, with adequate resolution of the migrating sedimentation boundary and, where relevant, the bottom back-diffusion region. Computational efficiency is therefore important, particularly because sedimentation-velocity analysis can require repeated solution of the Lamm equation during parameter estimation and distribution analysis. A dual-boundary-tracking finite element method (DBTFEM) is developed that combines a fixed, physically informed nonuniform radial grid with prediction-assisted dynamic selection of active nodes. The migrating sedimentation-boundary and bottom regions are retained as two coupled active blocks, while concentrations in the masked solution plateau are reconstructed analytically. This formulation concentrates the numerical solve on dynamically relevant regions of the radial domain while retaining the underlying finite element framework. The method is evaluated using six ideal, non-interacting sedimentation cases spanning different transport regimes and is compared with high-resolution numerical reference solutions. DBTFEM reproduces the interior concentration profiles accurately while substantially reducing the number of active spatial degrees of freedom in cases where the sedimentation and bottom regions remain sufficiently separated. Separate comparisons of grid allocation and active-node masking show that the benefit of the nonuniform grid is profile dependent, with the largest improvements obtained for steep sedimentation boundaries, whereas broader profiles exhibit accuracy comparable to that obtained with conventional uniform grids. Overall, DBTFEM provides an efficient finite element framework for finite-column Lamm equation simulation by combining physically informed spatial resolution with dynamic reduction of the active solution space.
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