and it can be extended to identify the allosteric pathways and detect
potential novel allosteric sites [10–12].
In the next two subsections we give a brief review of how to
predict flexibility of proteins with rigidity theory methods, focusing
on method FIRST [13]. In Subheading 2 we discuss rigidity-based
prediction of allosteric communication using the RTA analysis.
1.1 Analyzing
Molecular/Protein
Rigidity
with Mathematical
Rigidity Theory
To understand how proteins function including allosteric transmission requires deep knowledge of protein flexibility and its dynamics.
Protein motions take place on a wide range of time scales from rapid
bond-vibrations on the femtosecond range to large-amplitude collective motions occurring on milliseconds-seconds range
[14, 15]. A typical protein can contain thousands of conformational degrees of freedom, whose conformational fluctuations
among the structural ensemble members are rapid, transient, and
result in structures that are mostly spectroscopically undistinguishable compared to the ground state [14–17]. The ultimate desire is
to observe proteins move in real time at atomistic level as they
accomplish their function, but despite many advances in experimental measures of dynamics and biophysical and computational
methods including molecular dynamic simulations we are still far
from actualizing this goal [14, 17]. The computational time
needed to investigate large-scale functionally relevant motions
including those of allosteric transmissions with MD simulations,
even with special-purpose commodity computer clusters such as
Anton, is beyond practical wide-range applications [18]. To tackle
this challenge, there is a clear need to come up with alternate and
fast computational methods that simplify the force fields which can
still provide accurate and efficient protein flexibility predictions that
are in agreement with experimental measures. Numerous advances
in the mathematical rigidity theory [6–8, 19, 20] over the last
35 years have facilitated developments of several emerging technologies [8, 10–12, 16, 21–24] for fast computational predictions of
both protein flexibility and their dynamics.
Rigidity theory examines the rigidity/flexibility of frameworks
which are specified by geometric constraints (distances, directions,
etc.) on a collection of points and rigid bodies [6, 7] which has
many applications to both natural structures (molecules, crystals,
etc.) and engineered structures (bridges, robots, etc.) [13, 16, 22,
23, 25]. Proteins are modeled as constrained geometric molecular
frameworks (a mechanical linkage in kinematics and robotics vocabulary) consisting of atoms and an assortment of linking intermolecular forces (constraints) [7]. In a molecular framework model of a
protein (in rigidity theory referred to as a body-bar framework
[7, 26] (see Fig. 1a) we assume the angles between the bonds of
an atom (body) are fixed allowing dihedral angles to freely rotate,
Probing Allosteric Mechanism with Rigidity Transmission
63
potential novel allosteric sites [10–12].
In the next two subsections we give a brief review of how to
predict flexibility of proteins with rigidity theory methods, focusing
on method FIRST [13]. In Subheading 2 we discuss rigidity-based
prediction of allosteric communication using the RTA analysis.
1.1 Analyzing
Molecular/Protein
Rigidity
with Mathematical
Rigidity Theory
To understand how proteins function including allosteric transmission requires deep knowledge of protein flexibility and its dynamics.
Protein motions take place on a wide range of time scales from rapid
bond-vibrations on the femtosecond range to large-amplitude collective motions occurring on milliseconds-seconds range
[14, 15]. A typical protein can contain thousands of conformational degrees of freedom, whose conformational fluctuations
among the structural ensemble members are rapid, transient, and
result in structures that are mostly spectroscopically undistinguishable compared to the ground state [14–17]. The ultimate desire is
to observe proteins move in real time at atomistic level as they
accomplish their function, but despite many advances in experimental measures of dynamics and biophysical and computational
methods including molecular dynamic simulations we are still far
from actualizing this goal [14, 17]. The computational time
needed to investigate large-scale functionally relevant motions
including those of allosteric transmissions with MD simulations,
even with special-purpose commodity computer clusters such as
Anton, is beyond practical wide-range applications [18]. To tackle
this challenge, there is a clear need to come up with alternate and
fast computational methods that simplify the force fields which can
still provide accurate and efficient protein flexibility predictions that
are in agreement with experimental measures. Numerous advances
in the mathematical rigidity theory [6–8, 19, 20] over the last
35 years have facilitated developments of several emerging technologies [8, 10–12, 16, 21–24] for fast computational predictions of
both protein flexibility and their dynamics.
Rigidity theory examines the rigidity/flexibility of frameworks
which are specified by geometric constraints (distances, directions,
etc.) on a collection of points and rigid bodies [6, 7] which has
many applications to both natural structures (molecules, crystals,
etc.) and engineered structures (bridges, robots, etc.) [13, 16, 22,
23, 25]. Proteins are modeled as constrained geometric molecular
frameworks (a mechanical linkage in kinematics and robotics vocabulary) consisting of atoms and an assortment of linking intermolecular forces (constraints) [7]. In a molecular framework model of a
protein (in rigidity theory referred to as a body-bar framework
[7, 26] (see Fig. 1a) we assume the angles between the bonds of
an atom (body) are fixed allowing dihedral angles to freely rotate,
Probing Allosteric Mechanism with Rigidity Transmission
63