Using spermidine (SPD 3+ ), a trivalent polyamine, a similar nanotube-network transition was observed but at a higher concentration (5 mM, Figs. ). Interestingly, this concentration increase was larger than a simple compensation of valency as revealed by the larger charge ratio at the transition (ρ = 143, Fig. ). We compared the effect of these two polyamines to the simple dication Mg 2+ and found the same behaviour, at even a larger concentration (50 mM, Figs. ) and ρ (ρ = 1190, Figs. 3, S5), confirming not only the importance of valency in driving the process but also that such nanotube assembly mainly relies on electrostatic attraction rather than specific chemical interactions. For SPD 3+ , bundles were observed for higher concentrations and charge ratios but were smaller in size that with SPM 4+ while we could not detect bundles in the highest magnesium concentrations tested in our experiments. This further highlights the importance of a high condensation agent valency to get large and dense assemblies of DNA nanotubes. For a given valency, the evolution observed in all these phase diagrams is in agreement with the assembly of rigid negatively charged polyelectrolytes, such as actin filaments or microtubules, forming bundles in the presence of sufficiently high concentrations of multivalent cations, as shown in the past both experimentally and theoretically. The valency-dependence is also reminiscent to the compaction of double-stranded DNA by multivalent cations, where the neutralization of the DNA backbone through counter-ion condensation drives the process. According to the Manning-Oosawa picture, 36,37 counter-ions of valency z condense on the DNA backbone leading to an average charge neutralization θ = 1 -d/(zlB) where d is the average distance between DNA charges (0.17 nm) and lB is the Bjerrum length (lB = e 2 /(4πεkBT), with the e the elementary charge, ε the dielectric constant of the solvent, kB the Boltzmann constant and T the temperature). In water at 25 °C, lB = 0.7 nm and the neutralization only depends on z: θ = 1 -0.24/z. Therefore, adding multivalent cations to double-stranded DNA induces an entropically favourable counter-ion exchange and a progressive neutralization of DNA leading to its compaction when θ becomes too large (≈ 0.89). The higher the valency, the more entropically favorable the counter-ion exchange is, the more efficient DNA neutralization becomes, and the lower is the charge ratio necessary to induce DNA compaction. Notably, the same evolution was observed in the nanotube condensation diagram where transitions to higher-order structures were systematically observed at a lower ρ when z increased (Fig. ).