Mitochondria or evolved versions of them are found in all eukaryotes.

Based on extensive molecular analyses of their genomes, it is clear that they are derived from endosymbiotic alpha-proteobacteria.

The acquisition of mitochondria by the ancestral protoeukaryote occurred over 1.5 x 10⁹ years ago.

Since that time, mitochondria have been passed from cell to cell during the process of cell division.

Mitochondria have their own DNA and replicate by fission.

Over this period of time, many genes originally within mitochondria have moved to the nucleus of the cell.

In some organisms, such as Giardia intestinalis the mitochondria have changed so much as to be barely recognizable - nevertheless, it appears that all eukaryotes have mitochondria or mitochondrial derivatives.

Mitochondria can be recognized by the presence of a double membrane.

Based on the endosymbiotic theory, it appears that the outer membrane is derived from the original protoeukaryotic host, while the inner membrane is derived from the plasma membrane of the proteobacterial symbiont.

The other double-membraned organelle found in eukaryotes is the chloroplast, which also arose as a result of a endosymbiotic relationship, this time with a photosynthetic cyanobacterium.

In most eukaryotes, the main and best understood function of the mitochondria is to carry out respiration.

In the bacteria from which mitochondria were derived this process occurs across the plasma membrane.

In mitochondria is occurs across the inner membrane.

In both cases, the movement of high energy electrons through a membrane-embedded 'electron transport chain' is used to generate a [H⁺] gradient across the membrane.

The movement of H⁺s back through the membrane is mediated by the membrane-bound ATP synthase complex.

As H⁺ move through the ATP synthase, a part of the protein rotates and generates ATP from ADP and phosphate.

You might be interested in the "Extracting and Storing Energy" section of Biofundamentals

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