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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.

A eukaryote is an organism belonging to the lineage characterized by a membrane-bounded cell nucleus and extensive cellular compartmentalization. A nuclear envelope surrounds the nuclear genome; nuclear chromosomes are typically linear. Ribosomes perform biological translation outside the cell nucleus. See OpenStax Microbiology, 3.4: nuclear organization and protein synthesis. Specialized biological cells can lose their cell nucleus, as mature mammalian red blood cells do, while remaining parts of eukaryotic organisms; see OpenStax Biology 2e, 40.2: mature red blood cells.

Yeasts, plants and animals are all eukaryotes. Typical cells in these groups have mitochondria, but chloroplasts belong to photosynthetic plant and algae lineages, not to every eukaryote. A cell wall occurs in typical plant and fungus cells but is absent from animal cells. These are comparisons of representative cells, not a checklist every specialized cell must satisfy. See OpenStax Biology 2e, 4.3: nucleus, organelles and representative cells.

The interesting feature is controlled separation: cell membranes and organelle membranes let different cellular processes occupy distinct compartments while membrane transport connects them. The evolution of eukaryotes also involved partnership: mitochondria descend from a bacterial endosymbiosis. Modern phylogenetics supports an archaeal ancestry for the nuclear lineage, alongside this bacterial contribution; see two-domain system. Thus the history of a whole eukaryote cannot be read as the history of just one of its genes.

Mitosis allocates copied chromosomes to daughter cell nuclei. The spindle apparatus aligns sister chromatids before separation in anaphase.

The copying has already occurred during S phase, before mitosis. Nuclear chromosome segregation and cytokinesis, the partition of the cytoplasm, are distinct processes: two daughter cell nuclei can form inside one undivided biological cell.^[ref]

The original cartoon follows a hypothetical diploid animal biological cell with $2n=4$. Its two homologous chromosome pairs are $A,a$ and $B,b$; upper- and lower-case letters identify homologues, not dominant and recessive alleles. In prophase and metaphase, each replicated chromosome contains two sister chromatids: four replicated chromosomes contain eight chromatids, rather than eight replicated chromosomes. The X shape depicts two associated copies, not the shape of every chromosome throughout the cell cycle.

This open mitosis cartoon shows nuclear envelope breakdown and reassembly. Closed mitosis in some yeasts retains the nuclear envelope while daughter chromosomes separate inside it; the cartoon is therefore not a universal cellular architecture.^[ref]

The nuclear envelope is the double-membrane boundary around a cell nucleus. Regulated passage connects nuclear processes to the surrounding cytoplasm: transcription produces RNA in the cell nucleus, while exported messenger RNA can undergo biological translation on ribosomes outside it. See NHGRI: nuclear membrane.

A nuclear pore complex is an assembly of proteins spanning the nuclear envelope and regulating passage between the cell nucleus and cytosol. It is a selective gateway, not simply an unregulated hole: messenger RNA export and import of nuclear proteins require appropriate transport machinery. See Alberts et al.: nuclear pore complexes.

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.

In open mitosis, nuclear envelope breakdown permits spindle apparatus microtubules to reach kinetochores at centromere regions.

In open mitosis, telophase includes reassembly of nuclear envelopes around the separated chromosome groups and relaxation of their condensation.^[ref]