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Estimating how swimming speed changes as a function of chain length
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Chain formation can enhance the vertical migration of phytoplankton through turbulence

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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, $FP(n)$, scales linearly with the number of cells, n, such that $FP(n)=nFP(1)$, where $FP(1)$ 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, FD, can be modeled using Stokes law as $FD(n)=3πμVC(n)deK$, where μ is the dynamic fluid viscosity, $VC(n)$ is the swimming speed, K is a shape correction factor, and de is the equivalent diameter of the chain.

We considered two different models for FD, which differ in their formulation of K and de.The first model approximates a chain as a row of n rigidly attached spheres of diameter d, where de = n1/3d is the diameter of a single sphere whose volume is equal to the sum of the volume of all of the spheres within a chain and K is a correction factor that has been quantified via numerical simulations and with experiments (42, 43). Equating the drag force on the chain of spheres with the propulsive force yields $VC(n)=FP(1)n/3πμdeK$. This expression can be simplified by considering the swimming speed of a single cell of diameter d, which exerts a propulsive force $FP(1)$ that is balanced by a drag force of $FD=3πμVC(1)d$, which yields $FP(1)=3πμVC(1)d$. Substitution thus yields a chain swimming speed of $VC(n)=n2/3VC(1)/K$, and dividing both sides of this expression by the Kolmogorov velocity allows us to nondimensionalize, yielding Φ(n) = n2/3Φ(1)/K.

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 de is the diameter of the spheroid’s minor axis. Using the same process as above yields Φ(n) = nΦ(1)/K, where the correction factor K is given by (44)For convenience, we provide the numerical values of K for both models in table S1.

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