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Among the biological phenomena that fall within the emerging field of "quantum biology" is the suggestion that magnetically sensitive chemical reactions are responsible for the magnetic compass of migratory birds. It has been proposed that transient radical pairs are formed by photo-induced electron transfer reactions in cryptochrome proteins and that their coherent spin dynamics are influenced by the geomagnetic field leading to changes in the quantum yield of the signaling state of the protein. Despite a variety of supporting evidence, it is still not clear whether cryptochromes have the properties required to respond to magnetic interactions orders of magnitude weaker than the thermal energy, k(B)T. Here we demonstrate that the kinetics and quantum yields of photo-induced flavin-tryptophan radical pairs in cryptochrome are indeed magnetically sensitive. The mechanistic origin of the magnetic field effect is clarified, its dependence on the strength of the magnetic field measured, and the rates of relevant spin-dependent, spin-independent, and spin-decoherence processes determined. We argue that cryptochrome is fit for purpose as a chemical magnetoreceptor.