If ({mathcal {C}}) is a non-trivial cluster decomposition of ({1,ldots,N}), ({mathcal {C}}={C_1,ldots,C_k}) and (h(C_j)) is the internal Hamiltonian of (C_j), the set of (E_1+cdots +E_k) where (E_j) is an eigenvalue of (h(C_j)) is called the set of thresholds (if some (C_ell ) has one particle, then (h(C_ell )) is the zero operator on ({mathbb {C}}) and (E_ell =0)).
We found that the correspondence between the average rigid cluster decomposition of the PG to that predicted by the VPG is in markedly good agreement when using a uniform probability, p, for H-bonds to be present in the protein.
The algorithm is based on an adaptive process of cluster decomposition and combination and the process is iterated until the number of clusters becomes stable.
Furthermore, the rigid cluster decomposition is unique, and, all rigid clusters comprise of contiguous sets of nearest neighbor vertices.
In a previous study [ 29] the rigid cluster decomposition (RCD) from the VPG was compared to the ensemble-averaged RCD of the PG.
(Clustered Jacobi coordinates) Given ({1,ldots,N}), a cluster decomposition or clustering, ({mathcal {C}}= {C_ell }_{ell =1}^k), is a partition, i.e. a family of disjoint subsets whose union is ({1,ldots,N}).
BiNoM also implements a collection of powerful graph-based functions and algorithms such as path analysis, decomposition by involvement of an entity or cyclic decomposition, subnetworks clustering and decomposition of a large network in modules.
We have in prior work compared the detailed rigid cluster decompositions between the VPG and PG to a large dataset of 273 proteins [ 29].
Afterwards, these decomposed structures were consistently relabeled, clustered and transition pathways between clusters constructed: The decomposition of higher oligomers into dimeric structures was performed separately for trajectories which contained all trimeric, tetrameric and decameric aggregates observed in all simulations for both, PHF6 and IB12, peptides.
The catalyst solution concentration determines the uniformity and density of catalyst on the fibers, H2 exposure mediates formation of catalyst clusters, and thermal decomposition of the reactant mixture activates the catalyst particles to achieve uniform aligned growth.
The presence of this metal cluster allows the catalytic decomposition of hydrogen peroxide and the resulting oxygen bubbles trigger the propulsion of the object.
