Estimating how swimming speed changes as a function of chain length

This protocol is extracted from research article:

Chain formation can enhance the vertical migration of phytoplankton through turbulence

**
Sci Adv**,
Oct 16, 2019;
DOI:
10.1126/sciadv.aaw7879

Chain formation can enhance the vertical migration of phytoplankton through turbulence

Procedure

Phytoplankton chains have been widely reported to increase their swimming speed as the number of cells increases (*3*, *6*, *8*, *19*). This finding has been attributed to the fact that adding more cells to a chain increases its propulsive force more than it increases its hydrodynamic drag (*8*). While a regression of experimental data from the literature captures this process, we also considered two theoretical models (green lines in Fig. 4A).

Similar to a previous study (*8*), both of our models assume that the propulsive force of a chain, ${F}_{\mathrm{P}}^{(n)}$, scales linearly with the number of cells, *n*, such that ${F}_{\mathrm{P}}^{(n)}=n{F}_{\mathrm{P}}^{(1)}$, where ${F}_{\mathrm{P}}^{(1)}$ is the propulsive force generated by a single cell. This assumes that the cells within a chain do not interfere with one another’s propulsion within a chain; thus, this formulation may represent an upper bound (*8*). The drag force on a chain, *F*_{D}, can be modeled using Stokes law as ${F}_{\mathrm{D}}^{(n)}=3\mathrm{\pi}\mathrm{\mu}{V}_{\mathrm{C}}^{(n)}{d}_{e}K$, where μ is the dynamic fluid viscosity, ${V}_{\mathrm{C}}^{(n)}$ is the swimming speed, *K* is a shape correction factor, and *d _{e}* is the equivalent diameter of the chain.

We considered two different models for *F*_{D}, which differ in their formulation of *K* and *d _{e}*.The first model approximates a chain as a row of

The second model approximates a chain as a prolate spheroid whose aspect ratio is equal to the number of cells within the chain, *n*, and *d _{e}* is the diameter of the spheroid’s minor axis. Using the same process as above yields Φ

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