# Helical and oscillatory microswimmer motility statistics from differential dynamic microscopy

## TL;DR

This paper develops a theoretical framework to analyze the three-dimensional motion of micro-swimmers using Differential Dynamic Microscopy data, enabling extraction of motility parameters without advanced 3D imaging.

## Contribution

The authors derive ISF expressions for helical and oscillatory swimmers and demonstrate how to extract 3D motility parameters from 2D DDM data, simplifying experimental analysis.

## Key findings

- Derived ISF models for helical and oscillatory swimmers
- Validated the framework with simulated swimmer data
- Showed how to obtain 3D motility parameters from 2D DDM data

## Abstract

The experimental characterisation of the swimming statistics of populations of microorganisms or artificially propelled particles is essential for understanding the physics of active systems and their exploitation. Here, we construct a theoretical framework to extract information on the three-dimensional motion of micro-swimmers from the Intermediate Scattering Function (ISF) obtained from Differential Dynamic Microscopy (DDM). We derive theoretical expressions for the ISF of helical and oscillatory breaststroke swimmers, and test the theoretical framework by applying it to video sequences generated from simulated swimmers with precisely-controlled dynamics. We then discuss how our theory can be applied to the experimental study of helical swimmers, such as active Janus colloids or suspensions of motile microalgae. In particular, we show how fitting DDM data to a simple, non-helical ISF model can be used to derive three-dimensional helical motility parameters, which can therefore be obtained without specialised 3D microscopy equipment. Finally, we discus how our results aid the study of active matter and describe applications of biological and ecological importance.