Carbon nanotubes coated with alginic acid (AA) through noncovalent functionalization have been shown to be soluble and dispersed in water. In the present contribution, all-atom molecular dynamics simulations have been performed to probe the self-assembly mechanism that underlies the formation of complexes by AA and a single-walled carbon nanotube (SWCNT), both in the gas phase and in an aqueous solution. Results of these simulations reveal that AA can wrap around SWCNT by virtue of van der Waals attractions and organize into a compact helical structure, a process induced in the gas phase by hydrogen-bonding interactions. In contrast, in an alginate aqueous solution, a loose helical wrapping mode is found to be favored by virtue of electrostatic repulsions in conjunction with the weakening of hydrogen-bonding interactions. Documented experimentally (Liu, Y.; et al. Small 2006, 2, 874-878) and coined "Great Wall of China" motif, the typical arrangement of AA residues around the tubular structure, conducive to dissolve nanotubes, is observed in the present simulations. Investigation of metal cations binding to AA suggests that calcium ions can mediate aggregation of AA chains by interacting strongly with the carboxylate groups, thereby leading to reverse unwrapping. The results reported in this work shed meaningful light on the potential of noncovalent functionalization for solubilizing carbon nanotubes, and open exciting perspectives for the design of new wrapping agents that are envisioned to form the basis of innovative nanomaterials targeted at chemical and biomedical applications.