1.2 Protein Rigidity/
Flexibility Analysis
with Method FIRST
Given a PDB structure or an ensemble of structures, the program
FIRST [13] (and its various spinoff methods—Kinari, DCA, CNA
and others [20]) converts the structure to a body-bar multigraph
(network) model of a protein, consisting of vertices (atoms) and
edges (covalent bonds, hydrogen bonds, hydrophobic contacts,
and electrostatic interactions) (Fig. 1d, e). The strength of each
hydrogen bond is calculated using an energy potential [13]. A user
selects a hydrogen bond energy cutoff value such that all bonds
weaker than this cutoff are ignored and the final constraint bodybar multigraph is obtained. FIRST then applies the pebble game
algorithm [8, 27] on the multigraph which checks the combinatorial characterization of rigidity prescribed in the molecular theorem
[7, 19]. The pebble game determines if each constraint (bar/edge)
is “independent” (i.e., removes a DOF from the network) or is
otherwise “redundant.” Pebbles are synonymous with conformational degrees of freedom and a removal of a pebble indicates the
inserted constraint (edge) is independent. The pebble game finally
decomposes the protein into rigid clusters and flexible regions. A
rigid cluster moves as a single rigid body with its trivial 6 DOF
(a combination of 3 rotations and 3 translations). A typical protein
normally consists of several rigid regions connected by flexible
linkers (Fig. 2). Given such a decomposition of a protein into
rigid and flexible connections, fast Monte-Carlo methods such as
FRODA [23] (which give 100,000 speedups compared to MD
simulation) go a step further and can simulate the actual protein
motions and explore their dynamics. Sampling of conformational
space and dynamics can be done on very large systems, such as
ribosome and even viral capsids [28], and we have recently been
extending these techniques and applied it on intrinsically flexible
proteins which have a substantial amount of disorder and an
extremely high number of internal DOF [24].
In Fig. 2a–d we have shown the output of FIRST on a human
adenosine A 2A GPCR at several hydrogen bond energy cutoffs.
Hydrogen bonds can be removed one by one (i.e., by lowering of
hydrogen bond energy cutoff) in the order of increasing strength,
while maintaining all other covalent and hydrophobic interactions
intact, and then repeating the analysis as hydrogen bonds are
removed while recalculating rigid and flexible regions. Change in
rigidity can be visualized in the hydrogen bond “dilution plot”
(Fig. 2e). FIRST can predict the rigid clusters and flexible connections (known as the rigid cluster decomposition) in less than a second
on a standard PC/laptop. Many studies have demonstrated that
FIRST gives accurate predictions of flexibility and rigidity in proteins that are in agreement with experiments [10, 11, 16, 22].
Probing Allosteric Mechanism with Rigidity Transmission
65
Flexibility Analysis
with Method FIRST
Given a PDB structure or an ensemble of structures, the program
FIRST [13] (and its various spinoff methods—Kinari, DCA, CNA
and others [20]) converts the structure to a body-bar multigraph
(network) model of a protein, consisting of vertices (atoms) and
edges (covalent bonds, hydrogen bonds, hydrophobic contacts,
and electrostatic interactions) (Fig. 1d, e). The strength of each
hydrogen bond is calculated using an energy potential [13]. A user
selects a hydrogen bond energy cutoff value such that all bonds
weaker than this cutoff are ignored and the final constraint bodybar multigraph is obtained. FIRST then applies the pebble game
algorithm [8, 27] on the multigraph which checks the combinatorial characterization of rigidity prescribed in the molecular theorem
[7, 19]. The pebble game determines if each constraint (bar/edge)
is “independent” (i.e., removes a DOF from the network) or is
otherwise “redundant.” Pebbles are synonymous with conformational degrees of freedom and a removal of a pebble indicates the
inserted constraint (edge) is independent. The pebble game finally
decomposes the protein into rigid clusters and flexible regions. A
rigid cluster moves as a single rigid body with its trivial 6 DOF
(a combination of 3 rotations and 3 translations). A typical protein
normally consists of several rigid regions connected by flexible
linkers (Fig. 2). Given such a decomposition of a protein into
rigid and flexible connections, fast Monte-Carlo methods such as
FRODA [23] (which give 100,000 speedups compared to MD
simulation) go a step further and can simulate the actual protein
motions and explore their dynamics. Sampling of conformational
space and dynamics can be done on very large systems, such as
ribosome and even viral capsids [28], and we have recently been
extending these techniques and applied it on intrinsically flexible
proteins which have a substantial amount of disorder and an
extremely high number of internal DOF [24].
In Fig. 2a–d we have shown the output of FIRST on a human
adenosine A 2A GPCR at several hydrogen bond energy cutoffs.
Hydrogen bonds can be removed one by one (i.e., by lowering of
hydrogen bond energy cutoff) in the order of increasing strength,
while maintaining all other covalent and hydrophobic interactions
intact, and then repeating the analysis as hydrogen bonds are
removed while recalculating rigid and flexible regions. Change in
rigidity can be visualized in the hydrogen bond “dilution plot”
(Fig. 2e). FIRST can predict the rigid clusters and flexible connections (known as the rigid cluster decomposition) in less than a second
on a standard PC/laptop. Many studies have demonstrated that
FIRST gives accurate predictions of flexibility and rigidity in proteins that are in agreement with experiments [10, 11, 16, 22].
Probing Allosteric Mechanism with Rigidity Transmission
65