The ability of some animals, most notably migratory birds, to sense magnetic fields is still poorly understood. It has been suggested that this "magnetic sense" may be mediated by the blue light receptor protein cryptochrome, which is known to be localized in the retinas of migratory birds. Cryptochromes are a class of photoreceptor signaling proteins that are found in a wide variety of organisms and that primarily perform regulatory functions, such as the entrainment of circadian rhythm in mammals and the inhibition of hypocotyl growth in plants. Recent experiments have shown that the activity of cryptochrome-1 in Arabidopsis thaliana is enhanced by the presence of a weak external magnetic field, confirming the ability of cryptochrome to mediate magnetic field responses. Cryptochrome's signaling is tied to the photoreduction of an internally bound chromophore, flavin adenine dinucleotide. The spin chemistry of this photoreduction process, which involves electron transfer from a chain of three tryptophans, can be modulated by the presence of a magnetic field in an effect known as the radical-pair mechanism. Here we present and analyze a model of the flavin-adenine-dinucleotide-tryptophan chain system that incorporates realistic hyperfine coupling constants and reaction rate constants. Our calculations show that the radical-pair mechanism in cryptochrome can produce an increase in the protein's signaling activity of approximately 10% for magnetic fields on the order of 5 G, which is consistent with experimental results. These calculations, in view of the similarity between bird and plant cryptochromes, provide further support for a cryptochrome-based model of avian magnetoreception.