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Cristae (singular crista) are the internal compartments formed by the inner membrane of a mitochondrion. They are studded with proteins, including ATP synthase and a variety of cytochromes. The maximum surface for chemical reactions to occur is within the mitochondria. This allows cellular respiration (aerobic respiration since the mitochondrion requires oxygen) to occur.
Electron transport chain of the cristae 
NADH is oxidized into NAD+, H+ ions, and electrons by an enzyme. FADH2 is also oxidized into H+ ions, electrons, and FAD. As those electrons travel further through the electron transport chain in the inner membrane, energy is gradually released and used to pump the hydrogen ions from the splitting of NADH and FADH2 into the space between the inner membrane and the outer membrane (called the intermembrane space), creating an electrochemical gradient. This electrochemical gradient creates potential energy across the inner mitochondrial membrane, known as the proton-motive force. As a result, chemiosmosis occurs, producing ATP from ADP and a phosphate group when ATP synthase harnesses the potential energy from the concentration gradient formed by the amount of H+ ions. H+ ions passively pass into the mitochondrian matrix by the ATP synthase, and later on help to reform H2O.
The electron transport chain requires a constant supply of electrons in order to properly function and generate ATP. However, the electrons that have entered the electron transport chain would eventually pile up like cars traveling down a one-way dead-end street. Those electrons are finally accepted by oxygen (O2), which combine with some of the hydrogen ions from the mitochondrian matrix through ATP synthase and the electrons that had traveled through the electron transport chain. As a result they form two molecules of water (H2O). By accepting the electrons, oxygen allows the electron transport chain to continue functioning.
The electrons from each NADH molecule can form a total of 3 ATPs from ADPs and phosphate groups through the electron transport chain, while each FADH2 molecule can produce a total of 2 ATPs. As a result, the 10 NADH molecules (from glycolysis and the Krebs cycle) and the 2 FADH2 molecules can form a total of 34 ATPs from this electron transport chain during aerobic respiration. This means that combined with the Krebs Cycle and glycolysis, the efficiency for the electron transport chain is about 65%, as compared to only 3.5% efficiency for glycolysis alone.
The cristae greatly increase the surface area on which the above mentioned reactions take place. If they were absent, the inner membrane would (perhaps) be reduced to a single spherical shape, and with less reaction surface available, the reaction efficiency would be likewise reduced. Therefore, cristae are a necessity for the mitochondria to function efficiently.
One mathematical modeling study has suggested that the optical properties of the cristae in filamentous mitochondria may affect the generation and propagation of light within the tissue.
- Thar,R. and M.Kühl (2004). “Propagation of electromagetic radiation in mitochondria?”. J.Theoretical Biology, 230(2), 261-270.