Uniform asymptotic approximation of diffusion to a small target

Area of Science:

• Mathematical Biology
• Physical Chemistry
• Computational Science

Background:

• Predicting diffusion-driven target localization is crucial in biological modeling.
• Existing methods face challenges in accurately calculating first passage times in large domains.

Purpose of the Study:

• To develop accurate, uniform in time asymptotic expansions for first passage time density.
• To provide a robust method for analyzing diffusion to a small spherical target in bounded domains.

Main Methods:

• Combining long-time approximations (eigenvalue/eigenfunction of Laplacian) with short-time corrections (pseudopotential approximation).
• Developing uniform in time asymptotic expansions in target radius.
• Calculating expansions of the first passage time density and related statistics.

Main Results:

• The developed method accurately approximates the first passage time density.
• Demonstrated accuracy for a spherically symmetric problem with a concentric sphere domain.
• Provides accurate statistics for diffusion-driven target localization.

Conclusions:

• The combined asymptotic expansion method offers a powerful tool for diffusion-reaction problems.
• This approach enhances the understanding of molecular localization in biological systems.
• The method is validated for a key model system, paving the way for broader applications.