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Resolving how the distance between a chain’s center of mass and center of buoyancy varies with chain length
This protocol is extracted from research article:
Chain formation can enhance the vertical migration of phytoplankton through turbulence

Procedure

In this section, we used a simple model to show that h, the distance between a chain’s center of buoyancy, $zcb(n)$, and center of mass, $zcm(n)$, is the same as that of a solitary phytoplankton cell and does not vary with chain length, n. The center of buoyancy of a spherical individual cell (n = 1) occurs at its geometrical center such that $zcb(1)=d/2$, while h scales with cell diameter d (21) such that $h(1)=zcm(1)−zcb(1)=αd$. Therefore, via substitution, we obtained for a single cell that$zcm(1)=αd+d/2$To find the center of mass of a chain, we summed the product of the mass m and zcm for each of its constituent cells and then divided by the total mass of the chain. For a chain of two cells (fig. S5), we therefore have$zcm(2)=zcm(1)m+(d+zcm(1))m2m=zcm(1)+d/2$By definition, the separation between the center of mass and center of buoyancy for a two-cell chain is given by$h(2)=zcm(2)−zcb(2)$Noting that $zcb(2)=d$ by symmetry and substituting the previous result for $zcm(1)$, we obtained$h(2)=αd$which shows that a single cell and a two-cell chain have the same h.

Using the same logic, we then expanded this analysis for a chain of generic length n, which yields$zcm(n)=nzcm(1)+∑i=1n−1idn=zcm(1)+d(n−1)/2$and consequently$h(n)=zcm(n)−zcb(n)=zcm(1)+d(n−1)/2−nd/2=αd$Thus, a chain has the same h as each of its constituent cells.

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