|MadSci Network: Evolution|
What at first appeared to be a pretty straightforward answer has turned out to be quite complex. The simple answer was going to be: because the evolution of fur made the pigmentation of the skin irrelevant, such that the abilities to produce some of the ancestral pigments were lost to the mammals. This is mostly true, but the exceptions have proven to be as interesting as the answer itself.
In most vertebrates, color, or pigmentation, is produced by special pigment cells called chromatophores (chromato means "color"; phore means "producer") that lie between the inner (dermis) and outer (epidermis) layers of skin. In fish, amphibians, and reptiles these chromatophores come in three flavors: melanophores full of melanins, which are brown/black; xanthophores full of pteridines and carotenoids, which are yellow/red; and iridophores with special "reflecting platelets" that can be silvery to iridescent or complement the other pigments to produce any number of other colors. [The first two references at the bottom of the page give excellent descriptions of the roles of each chromatophore in the pigmentation of frogs and fish, respectively.]
In mammalian skin and hair, there is only one type of chromatophore, the melanophore, which is instead called the melanocyte. With only the one source of pigments for coat color, mammalian have evolved some interesting tricks to extend the spectrum of available colors. First, melanocytes can produce two distinct pigments: eumelanin, which is the traditional brown/black; and pheomelanin, which is yellow/red. Second, melanocytes deposit their pigments into the cells of growing hairs and can cycle through different pigmentation programs, such that a single hair can have stripes, as in the agouti coat color of many mammals. Third, the almost crystalline nature of the keratin arrangements in mammalian hair give fur excellent light-scattering properties. Altogether, these traits fill most of the gaps created by the loss of the other two chromatophores, such that there are white polar bears, yellow cats, red pandas, and green monkeys! (Well, they're not really green, in the way that you mean: they're more olive drab in an agouti sort of way.)
So, the summary would be: as early mammals developed thicker coats of hair, the mammalian chromatoblasts lost the ability to differentiate into xanthophores or iridophores and the surviving melanocytes (nee –phores) were left to make up the difference with their limited repertoire of colors to choose from – hence, no green. I said this would be the summary, because of one particular wrench that requires reassessing the list of available colors: many mammals have brilliantly colored eyes, using the same cells and pigments to color the iris that are used to color the skin and hair. [Discussed in the last reference below.] So, mammals can produce green colorations, but only in their eyes, apparently because the adaptive or courtship value of eye coloration was sufficiently distinct from that of coat color for each to have evolved separately, though both rely on the same cellular palette.
Ichikawa Y, Ohtani H, and Miura I (2001), "The yellow mutation in the frog Rana rugosa: pigment organelle deformities in the three types of chromatophore," Pigment Cell Res. 14(4):283-288
Quigley IK and Parichy DM (2002), "Pigment pattern formation in zebrafish: a model for developmental genetics and the evolution of form," Microsc Res Tech. 58(6):442-455
Oliphant LW, Hudon J, and Bagnara JT (1992), "Pigment cell refugia in homeotherms--the unique evolutionary position of the iris," Pigment Cell Res. 5(6):367-371
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