Determining the Gaussian probability distribution of the best-fit ellipsoid of revolution for a polymer chain from planar projections

This paper introduces an algorithm to obtain the probability distribution of the best-fit ellipsoid of an individual random-walk polymer chain from planar projections. This algorithm allows the true three-dimensional behaviour of polymer chains to be inferred from planar experimental measurements such as those obtained by light microscopy. Our algorithm is effective in retrieving the Gaussian probability density functions (PDFs) of evolving best-fit prolate and oblate ellipsoids with axial symmetry. The implementation depends on the properties of the projections of rigid ellipsoids from which the desired PDFs can be retrieved by optimization. Detailed derivation of the relevant formulae is provided, and examples are given to help to explain the algorithm. An error analysis shows that the algorithm has good properties of convergence and robustness. Therefore, the proposed algorithm is applicable to the analysis of data obtained from experiments.