Modeling Macromolecular Complexes: A Journey Across Scales - Modeling in Computational Biology and Biomedicine - page 35

a b c

80 4

Y-complex

λ =1

Number of complexes
Volume ratio curve
Target stoich. i.e. 16

70

60 3

50

40 2

30

20 1

10

0 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

λ =0

lambda

Fig. 1.15 Global assessment for the Y -complex. ( a ) The Hasse diagram representing the evolution
of the connected components. Fat nodes correspond to isolated copies. ( b ) Evolution of the number
of complexes and volume ratio r λ as a function of λ .( c ) The complex corresponding to the red fat
node of the Hasse diagram presented in ( a )

connected components. These components correspond to the so-called rings of the
NPC, whose structure is still under controversy. In fact, analysing the contacts
accounting for the closure of the two rings provides useful information, and is
currently being used to discuss hypotheses about the structure of these two rings.

Local assessment w.r.t. a 3D model. Assume now that we wish to compare
a complex C against a model T , which may come from a crystal structure or
which may have been designed in-silico—such as the model for the Y -complex
of Fig. 1.8 b. Assume that the model T comes with the pairwise contacts between its
constituting proteins. As discussed in Sect. 1.3.3 , we maintain the skeleton graph of
C , which precisely encodes the contacts between the toleranced proteins of C . Thus,
comparing C against its model T boils down to comparing two graphs. The two
standard operations to do so consist of computing the Maximal Common Induced
Sub-graph and the Maximal Common Edge Sub-graph of the two graphs [ 15 ].
These operations are used to compare the contacts encoded in complexes of the
toleranced model against those present in putative models. In particular, they have
allowed the design of a new 3D template for the T -complex [ 13 ].

1.4 Outlook

While investigating methodological developments for docking [ 20 ], Michael Con-
nolly claimed “ Geometry is not everything but it is the most fundamental thing ”
Indeed, geometrical reasoning applies everywhere in Biology: in formulating the
equations used to model the basic physical forces between atoms, in representing the
shapes of the macromolecules themselves, in describing their interacting surfaces,
and modeling of the structure of complexes, both small and large. Many of these ap-
proaches rely on simple models of the macromolecule, be it a van der Waals model,
a Voronoı diagram, or a low-resolution assembly of balls representing pseudo-
atoms. But geometric reasoning further allows one to build strong statements