Geometry and Topology of Folding Landscapes - Biological Networks - page 138

7.2.4. Rigid block models

One of the advantages of ENMs over more detailed force fields and simulations
is their ability to produce analytical results with a coarse-grained model. One can
quickly calculate the dynamics of a moderately large macromolecule simply by
building and diagonalizing its Hessian matrix, an O( N 3 ) computation.
Occasionally, though, the memory requirements of the calculation overwhelm
contemporary computers, and further coarse-graining is necessary. For example,
if one wishes to consider the motion of side chains, a Hessian based on α-carbon
coordinates alone is insufficient, and an ENM using all atoms must be used
instead. Such refinement increases the system size approximately tenfold,
possibly beyond the tolerance of available computational resources. Similarly,
very large molecular assemblies such as viral capsids can contain upwards of 10 5
residues, and their Hessian matrices cannot be easily handled by conventional
computing resources.
One way to overcome the problem of excessive system size is to bundle
several elements of the physical system into a single node. This method
faithfully reproduces the global dynamics of the system (Doruker et al. , 2000),
but does not produce detailed motions for all of the original nodes. Mixed
models (Kurkcuoglu et al. , 2003) that rely on multiple levels of coarse-graining
can provide detailed results only for specific regions of interest. A good method
for surmounting the problem of finding the normal modes of very large systems
is the rotations and translations of blocks (RTB) (Tama et al. , 2000), also called
block normal mode (BNM) (Li & Cui, 2002) method. This method assumes that
the system is constructed of n b rigid blocks, and that the normal modes of the
system can be expressed as rigid body rotations and translations of its constituent
blocks. Each block has 6 degrees of freedom, and the approximation reduces the
size of the system from 3 N to 6 n b .
Consider a system of N nodes that can be collected into n b < N rigid blocks
connected by elastic springs. As before, the generalized mass-weighted
coordinates form the 3 N -vector r , and the system's Hessian, H , is calculated
from the topology of the elastic network. We define the 3 N × 6 n b projection
matrix, P , from the 3 N -dimensional space of all nodes into the 6 n b -dimensional
space of rotations and translations of the rigid blocks. The relationship between
linear motion of a rigid body and the motion of its constituent components is
captured by the conservation of linear momentum in mass-weighted coordinates:

= ∑

ɺ ɺ ,

M r m r (7.24)

CM k k

k