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Algae is a broad name for several photosynthetic eukaryotic groups, rather than one exclusive clade. The older name blue-green algae for cyanobacteria is misleading in a comparison of cellular organization: cyanobacteria are bacteria, with neither chloroplasts nor a cell nucleus.

Archaea are a major evolutionary lineage distinct from bacteria. The cellular comparison here concerns the prokaryotic members of archaea; see two-domain system for broader clade usage. Their biological cells have a nucleoid rather than a membrane-bounded cell nucleus, yet aspects of DNA replication, transcription and biological translation resemble those of eukaryotes. Their characteristic lipids differ chemically from the usual bacterial membrane lipids, and their cell walls, when present, lack peptidoglycan. See OpenStax Biology 2e, 22.2: archaeal distinctions.

Some archaea perform methanogenesis, producing methane as part of their metabolism. Archaea also inhabit ordinary soils and oceans: extreme habitats do not define the lineage. See OpenStax Biology 2e, 22.1: moderate and extreme environments. Woese and Fox's 1977 analysis of ribosomal RNA distinguished the methanogenic group they called archaebacteria from typical bacteria. Their proposed three primary lines of descent were a historical interpretation, rather than the final resolution of deep phylogenetics. See Woese and Fox, 1977: Phylogenetic structure of the prokaryotic domain.

A clade comprises an ancestor and all its descendants. Membership is about common descent, not a taxonomic rank or a degree of cellular complexity. In a two-domain system, the nuclear lineage of eukaryotes nests within archaea; a group containing only the prokaryotic members excludes descendants and is therefore not that complete clade. See OpenStax Biology 2e, 20.2: clades and shared ancestry.

A prokaryote is an organism whose biological cells lack a membrane-bounded cell nucleus. The DNA of its chromosomes occupies a nucleoid within the cytoplasm. Bacteria and archaea have this cellular organization. Both contain ribosomes that perform biological translation and a selective cell membrane; neither is a bag of unorganized chemicals. See OpenStax Biology 2e, 4.2: prokaryotic organization.

The word prokaryotic describes organization, not a single exclusive clade. Grouping bacteria and archaea by the absence of a cell nucleus is useful for comparing biological cells, but does not establish that they share a more recent ancestor with one another than either does with eukaryotes. See three-domain system and two-domain system for the different ancestry models.

Prokaryotic does not mean membrane-free. Cyanobacteria contain internal thylakoid membranes for oxygenic photosynthesis, despite lacking chloroplasts and a cell nucleus. Many prokaryotes have circular chromosomes, but circular shape and a single chromosome are not defining requirements. See OpenStax Biology 2e, 22.2: bacterial and archaeal structures.

Taxonomy names and classifies organisms. Classification can organize known diversity and make comparisons useful, but ranks and names are not themselves evidence of ancestry. Phylogenetics infers that ancestry; a clade includes an ancestor and all its descendants. A shared feature such as lacking a cell nucleus need not identify an exclusive clade. The organism branch here organizes cell types for reading and is not a proposed phylogenetic tree. See OpenStax Biology 2e, 20.1: classification and evolutionary history.

The two-domain system recognizes bacteria and archaea as the two primary lineages, placing the nuclear ancestry of eukaryotes within archaea. In this model, archaea as a complete clade includes that eukaryotic lineage; restricting the name to prokaryotic members describes a different grouping. Williams and colleagues' study, published online in 2019 and in a 2020 journal issue, found support for this model and a close relationship to Asgard archaea. See Williams et al., 2020: Phylogenomics provides robust support for a two-domains tree of life.

This updates the interpretation of the 1977 and 1990 classifications rather than undoing their discovery of deep differences between bacteria and archaea. Deep phylogenetic trees depend on the sampled genes, organisms and inference model; the precise closest archaeal relatives and early sequence of events require further evidence. Bacterial ancestry of mitochondria through endosymbiosis is a separate part of eukaryotic history.

Yeasts are fungi whose growth is predominantly unicellular, commonly through budding or division into two cells. The word describes a growth form across different fungal lineages, rather than one species or an exclusive clade; some fungi can switch between unicellular and filamentous growth. Thus Saccharomyces cerevisiae is a yeast, but yeast is not another name for that one species. See Kurtzman, Fell and Boekhout, The Yeasts, chapter 1 (2011): definition and classification.

A yeast cell has eukaryotic organization, including a cell nucleus, mitochondria and a cell wall. Small size and a unicellular lifestyle do not make it a member of bacteria. These compartments let an experimentally accessible biological cell retain much of the organization shared with other eukaryotes. See Clark, Douglas and Choi, Biology 2e, 24.1: fungal cells and yeast growth.

A motivating experiment: can fermentation continue without intact yeast cells? In 1897, Eduard Buchner ground brewers' yeast with quartz sand and diatomite, added water, and pressed out a liquid extract. Adding concentrated cane sugar produced carbon dioxide after 15–60 minutes. A cell-retaining filter still yielded an active filtrate: with sterilized cane sugar in an icebox, fermentation began after about a day. Saturated lactose gave no observed fermentation. These are reported observations, not invented control results. See Buchner, Alcoholic fermentation without yeast cells (1897), translated excerpt hosted by Carmen Giunta.

The filtration comparison addresses the objection that surviving intact yeast cells caused the activity. The lactose comparison tests substrate dependence; it is not a no-sugar blank, nor a comparison at matched sugar concentrations. A proposed modern follow-up would also compare the same extract with and without added sugar, accounting for material already in the extract. That is an experimental suggestion, not a result attributed to Buchner. His report specifies brewers' yeast; it does not identify the preparation here as a pure Saccharomyces cerevisiae strain.

Heat treatment asks a different question: does destroying activity in the extract stop fermentation? Buchner's preliminary report described weak or absent activity after heating and filtration, rather than one clean universal threshold. Harden's 1914 account of the subsequent work summarized loss of activity on heating the juice to $50{}^{\circ }\mathrm{C}$. Harden also described requirements for both enzymes and coenzymes. Cell-free therefore means neither chemical-free nor a preparation containing one purified enzyme. See Arthur Harden, Alcoholic Fermentation (1914), chapters II and VI: cell-free activity and the fermenting complex.

The inference is that extracted cellular machinery can sustain these chemical reactions without an intact living biological cell. This does not show that the extract reproduces, grows or maintains itself like a yeast. Moreover, gas evolution alone does not identify ethanol: cellular respiration can also release carbon dioxide. Product identification is a separate measurement; Harden's account describes both ethanol and carbon dioxide as products of cell-free alcoholic fermentation. The beautiful separation is between the organism that produces and maintains the machinery and the machinery that catalyzes a particular transformation.