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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.

The cell nucleus is a membrane-bounded compartment characteristic of eukaryotic biological cells, containing most of the cellular DNA in a typical nucleated biological cell. Its double nuclear envelope separates this DNA from the surrounding cytoplasm. Most does not mean all: mitochondria and chloroplasts retain their own DNA. Nor does every specialized eukaryotic biological cell retain a cell nucleus: mature mammalian red blood cells have lost theirs. Prokaryotes organize their DNA in a nucleoid without a nuclear envelope, and still have ribosomes. See NHGRI: nucleus, ^[ref] and Clark, Douglas and Choi, Biology 2e, 40.2: mature mammalian red blood cells.

Follow one nuclear protein-coding gene: its DNA is used in transcription to make RNA; after the appropriate processing, messenger RNA is exported through a nuclear pore complex; a ribosome in the cytoplasm then uses it in biological translation to build an amino acid chain. The DNA remains in the cell nucleus. The RNA is a template, not the material converted into a protein. The ribosome diagram below follows this route; gene expression gives a complementary nucleotide sequence example. See Alberts et al.: selective RNA export and protein import.

This architecture makes gene expression interesting in a spatial way: nuclear transcription and cytoplasmic biological translation are separated by a selective boundary. A messenger RNA can be processed before export and use, while proteins needed for nuclear functions travel inward. Inside the cell nucleus, the cell nucleolus helps assemble ribosome subunits for export. Thus the compartment housing most DNA also helps build the machinery that interprets its messages elsewhere.

Robert Brown described and named the cell nucleus in his account of plant biological cells, read to the Linnean Society in 1831 and published in its Transactions in 1833. This historical observation identified a recurring structure, without establishing its later-understood role in DNA storage or gene expression. See Brown, On the Organs and Mode of Fecundation in Orchideæ and Asclepiadeæ, Transactions 16, 685–738 (1833). This is a documented contribution rather than a claim that no one had observed the structure earlier.

The cytoplasm is the material inside a cell membrane but outside the cell nucleus, when one is present. It includes the cytosol, the organelles suspended in it, and other structures; it is not just a name for the liquid. In a prokaryote, which has no cell nucleus, the nucleoid and ribosomes are within the cytoplasm. See Clark, Douglas and Choi, Biology 2e, 4.3: cytoplasm and cytosol and Clark, Douglas and Choi, Biology 2e, 4.2: prokaryotic organization.

Try classifying four positions in the diagram. A point in the pale blue region belongs to both cytosol and cytoplasm. A point inside the mitochondrion belongs to an organelle included in the cytoplasm, but not to the surrounding cytosol. A point on a free ribosome belongs to the cytoplasm, but the ribosome itself is not cytosol. A point inside the cell nucleus belongs to neither cytoplasm nor cytosol. These distinctions prevent the misleading inference that everything outside the cell nucleus shares one undivided liquid environment.

The usefulness of the cytoplasm is this combination of shared surroundings and specialized structures: enzymes in the cytosol carry out many steps of metabolism, while organelles maintain their own internal conditions. A soluble protein made by a free ribosome may work in the cytosol or be transported elsewhere, including into the cell nucleus. Cellular organization therefore combines separation with exchange, rather than assigning each protein permanently to its place of synthesis. See Alberts et al., Molecular Biology of the Cell (2002): traffic between nucleus and cytosol.

Escherichia coli, often abbreviated E. coli, is a bacterial species widely used to study DNA replication and gene expression. Its chromosome is in a nucleoid, and its ribosomes work in the cytoplasm, outside any cell nucleus. A result from one laboratory strain should not automatically be generalized to every member of the species.

The original 1997 report by Blattner and colleagues presented the K-12 MG1655 genome sequence as a circular chromosome of $4,639,221$ base pairs. See Blattner laboratory: original K-12 sequence and 1997 paper. As a bookkeeping example, one complete round of DNA replication from one such chromosome produces two daughter sequence copies totaling $2\times 4,639,221=9,278,442$ base pairs. This is twice the copied material, not twice the length of the reference genome; it does not specify the instantaneous DNA content of a rapidly growing E. coli cell.

A nucleoid is the region occupied by chromosome DNA in a prokaryotic biological cell. It has no surrounding nuclear envelope, but its DNA is organized rather than freely spread at random. See OpenStax Biology 2e, 4.2: the nucleoid.

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.