Meaning of Motif in Molecular Biology
In English the word, motif (borrowed from the French), has a variety of meanings in different areas. The one that is borrowed in molecular biology is that of pattern together with a hint, perhaps, of emblem or badge.
The word pattern indicates both repetition and a master mould from which copies are made. In molecular biology this indicates that this is not unique, it occurs repeatedly. The word emblem suggests a means of identifying the group to which something belongs. In molecular biology it is associated with a shared function for members of the group.
Types of Motif in Molecular Biology
Here the word motif is applied mainly to the three related macromolecules, DNA, RNA and protein. All of these are linear chains of restricted varieties of defined components (four nucleotides, 20 amino acids) arranged in defined ways which we refer to the sequence of the macromolecule. Within the overall sequence there can be sub-sequences, which, if they repeat represent patterns, and which may have functional significance. We refer to such patterns as:
- DNA sequence motifs
- RNA sequence motifs
- Protein (or amino acid)sequence motifs
The first of these is what D’haeseleer is referring to. It should be noted that these sequence motifs can be absolute, or consist of consensus sequences, such as the one for the ROX 1-binding site in the article cited:
However the nucleotides or amino acids are not the only components of macromolecules the arrangement of which can produce a pattern. Motifs in molecular biology can be composed of structural components. This is expressed in the following definition of protein motif:
Protein motifs are small regions of protein three-dimensional
structure or amino acid sequence shared among different proteins. They
are recognizable regions of protein structure that may (or may not) be
defined by a unique chemical or biological function.
And a similar definition that considers only structural motifs in proteins adds: “An example… is a helix-turn-helix motif.” This is a motif in certain proteins that bind DNA.
Identifying Sequence Motifs by Computer
The limit of the approach suggested is that statistically one would expect any small sequence pattern to recur in DNA, and the problem is how to tell which recurring patterns have biological significance. It should be realized that is not the sequence motif alone that makes it functional, but the context in which it is found. Thus, a TATA box or other DNA motifs that act as binding sites for transcription factors were discovered (and are differentiated from other random or non-functional occurrences in the genome) by their proximity to the positions where transcription is initiated. Their function was confirmed experimentally, e.g. by binding RNA polymerase to the DNA. (This, I hope, answers the last query about the function of motifs as protein-binding sites.)
So, in general, no. Although I admit that I have personally use computational approaches to discover new small hydrogen-bonded protein motifs (a somewhat specialized area).
The ferritin-like di-iron motif
This is not a DNA sequence motif. It is not even a protein sequence motif, but a structural protein motif. The definition can be found on InterPro:
This entry represents a group of proteins, containing ferritin-like
domain, which is an about 145-residue domain made of a four-helix
bundle surrounding a non-heme, non-sulphur, oxo-bridged diiron site.
The diiron site is contained within a twisted, left-handed four-
helix-bundle constituted of two anti-parallel helix pairs connected
through a left-handed crossover connection.
It is shown here with the helices coloured yellow and the iron atoms red spheres:

[From deMare et al. (1996)]
Footnotes
The helix–turn–helix is also referred to as a domain. The distinction between motif and domain is one of size (note the small in the definition of protein motif). However InterPro does refer to the ferritin-like di-iron pattern as a motif, so this can be regarded as acceptable usage.
One might argue that as the sequence and structure of proteins are specified in the DNA, the motif should also be in the DNA. This is a fallacy. The information is there, but in a cryptic form. The redundancy of the genetic code means that protein sequences are far more conserved than the corresponding DNA sequences, and the three-dimensional structural patterns are not evident from inspecting amino acid sequences.