Resolving how the distance between a chain’s center of mass and center of buoyancy varies with chain length

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 thatzcm(1)=αd+d/2To 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 havezcm(2)=zcm(1)m+(d+zcm(1))m2m=zcm(1)+d/2By definition, the separation between the center of mass and center of buoyancy for a two-cell chain is given byh(2)=zcm(2)zcb(2)Noting that zcb(2)=d by symmetry and substituting the previous result for zcm(1), we obtainedh(2)=αdwhich 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 yieldszcm(n)=nzcm(1)+i=1n1idn=zcm(1)+d(n1)/2and consequentlyh(n)=zcm(n)zcb(n)=zcm(1)+d(n1)/2nd/2=αdThus, a chain has the same h as each of its constituent cells.

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