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In common cladistic usage, a monophyletic group is a taxon (group of organisms) which forms a clade, meaning that it consists of a species and all its descendants. The term is synonymous with the uncommon term holophyly. Monophyletic groups are typically characterized by shared derived characteristics (synapomorphies).
Monophyly is contrasted with the terms paraphyly and polyphyly, which are most easily understood from the second diagram in this article. In current usage, a paraphyletic group consists of all of the descendants of a possibly hypothetical closest common ancestor minus one or more monophyletic groups (most usually one). A paraphyletic group is thus 'nearly' monophyletic (consistent with the meaning of the prefix 'para', namely 'near' or 'alongside'.) A polyphyletic group is any group other than a monophyletic group or a paraphyletic group, which like a paraphyletic group contains only some of the descendants of their closest common ancestor, but unlike a paraphyletic group is not characterized by the missing descendants forming one (or more) monophyletic groups.
These definitions have taken some time to be accepted. When the cladistic school of thought became mainstream in the 1960s, several alternative definitions were in use. Indeed, taxonomists sometimes used terms without defining them, leading to confusion in the early literature, a confusion which persists.
On the broadest scale, definitions fall into two groups.
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This incompatibility with the Linnaean and evolutionary taxonomy models led to an initial rift, not entirely healed, between the cladistic and more traditional schools of thought. Extreme cladists challenged the validity of Linnean taxa such as the Reptilia. Because birds, although descended from reptiles, are not themselves considered to be reptiles, cladists demanded that the taxon Reptilia be dismantled: a request that taxonomists were unwilling to heed. This stand-off was eventually resolved to a degree by the construction of the term 'paraphyletic' to describe closely related groups which included most but not all of the descendants of a common ancestor.
However, the coining of this term led to yet more confusion. Some scientists considered paraphyletic groups to be monophyletic (as they shared a common ancestor), where others insisted that monophyletic should continue to refer only to holophyletic groups. Another term, polyphyletic, fell outside of the definition of monophyly. A strict explanation of a paraphyletic group has not been published, but the consensus appears to be that paraphyletic groups consist of a monophyletic group, minus one or more smaller constituent clades – for instance the paraphyletic class Reptilia being "Amniotes minus birds and mammals". Polyphyletic groups can be thought of as a number of unrelated clades, for instance "warm blooded animals" = "birds plus mammals". Non-holophyletic groups are of little use for analysis of evolutionary processes, hence the calls for their "unnaming" - even though they are useful to scientists who are less concerned with the evolutionary past of groups. Naming is also a problem for monophyletic groups: because the number of ancestors from which to root monophyletic groups is almost infinite, giving each clade a unique name is impossible - as illustrated by the failed attempts to instigate a system called the Phylocode. Names obfuscate the really interesting part, which is the branching order, and are therefore of little utility to the cladist - at odds with the taxonomist, who since the time of Linnaeus has been naming species.
Intermediate, and particularly fossil, taxa can be considered to fall 'just outside' a widely accepted taxon. Examples are the very tetrapod-like fossil elpistostegalian fishes Panderichthys and Tiktaalik. The skull and upper finn structures are very similar to those of the first tetrapods, yet the limbs terminated in fins with fin-rays rather than feet with digits. The two are close to, but not members of the group consisting of four footed vertebrates. To reflect this phylogenetic proximity, they are termed termed 'stem group tetrapods' or simply 'stem tetrapods' - i.e. it lies on a branch more closely related to the lineage that led to tetrapods as recognised by a taxonomist than to their nearest living relatives, the lungfish. This concept closes the gap between taxonomy and cladistics at a broader scale, but is difficult to apply at a species-level resolution.