Eukaryotic is the characteristic most likely describes a mushroom but not Thermus aquaticus.
Any member of the eukaryotic group of organisms, which also includes the more well-known mushrooms and microbes like yeast and mold, is referred to as a fungus.
One of numerous thermophilic bacterial species that are a part of the Deinococcota phylum that can withstand high temperatures is Thermus aquaticus. Bacteria are classified as prokaryotes, which are unicellular life forms. Prokaryotes lack a membrane-bound nucleus and other internal components.
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