44
1
General Principles
where G P accounts for the favorable effects of mutual electrostatic polarization of the solute
and the solvent minus the cost of distorting the solvent, E internal accounts for the cost of
internal distortion of the solute and G CDS , which depends on the solvent-accessible surface
area, includes the effects of interactions dominated by the first solvation shell, such as cavitation [211,477,478]. Another approach is to use Langevin dynamics [333,479,480]. This area
is under active development [478,481,482,483,484,485].
Molecular dynamics allows the examination of the time dependence of a system that includes
a number of solute and solvent molecules in a cell [486,487,488]. The system of particles is
termed the ensemble and the number of particles, the volume and either the energy or the
temperature are kept constant. The time evolution of the ensemble is obtained from Newton’s
second law of motion:
dv
dt
=
F
m
(15)
where the acceleration is dv/dt, F is the force, and m is the atomic mass. Very small time steps
keep the energy constant. The force is obtained from the molecular mechanics force field,
dE pot
dr
= −F .
(16)
First derivatives of the potential energy with respect to the Cartesian coordinates are determined as part of the minimization process. The total energy of the system is the sum of the
potential and kinetic energies. The kinetic energy is proportional to the temperature:
E kinetic =
1
2
mv
2
=
3
2
kT
(17)
and this can be used to define the temperature of the ensemble. A difficulty is handling the borders of the cell and various approaches have been used, including periodic border conditions,
where the cell is symmetrically replicated in all directions.
The necessity of using small time steps means that many steps must be calculated in order
to adequately sample conformational space. Therefore, molecular dynamics calculations are
normally performed with class 1 force fields. Molecular dynamics simulations are analyzed by
following the trajectory of a variable with time, often angles for oligosaccharides. Most
modeling programs now can perform molecular dynamics calculations [347,462,466,489].
Quantum mechanical calculations [490,491,492,493] are now fast and accurate enough that
they have become the method of choice for studying conformations of individual monosaccharides and disaccharides [231,494,495,496,497]. The use of density functional theory [498]
allows the study of much larger systems. High-level methods have to be employed to yield
accurate results in terms of energies [499]. The continuing improvements in computational
power will result in acceleration of the use of these methods.
Acknowledgement
I would like to thank Al French for comments on this chapter from the previous edition.
1
General Principles
where G P accounts for the favorable effects of mutual electrostatic polarization of the solute
and the solvent minus the cost of distorting the solvent, E internal accounts for the cost of
internal distortion of the solute and G CDS , which depends on the solvent-accessible surface
area, includes the effects of interactions dominated by the first solvation shell, such as cavitation [211,477,478]. Another approach is to use Langevin dynamics [333,479,480]. This area
is under active development [478,481,482,483,484,485].
Molecular dynamics allows the examination of the time dependence of a system that includes
a number of solute and solvent molecules in a cell [486,487,488]. The system of particles is
termed the ensemble and the number of particles, the volume and either the energy or the
temperature are kept constant. The time evolution of the ensemble is obtained from Newton’s
second law of motion:
dv
dt
=
F
m
(15)
where the acceleration is dv/dt, F is the force, and m is the atomic mass. Very small time steps
keep the energy constant. The force is obtained from the molecular mechanics force field,
dE pot
dr
= −F .
(16)
First derivatives of the potential energy with respect to the Cartesian coordinates are determined as part of the minimization process. The total energy of the system is the sum of the
potential and kinetic energies. The kinetic energy is proportional to the temperature:
E kinetic =
1
2
mv
2
=
3
2
kT
(17)
and this can be used to define the temperature of the ensemble. A difficulty is handling the borders of the cell and various approaches have been used, including periodic border conditions,
where the cell is symmetrically replicated in all directions.
The necessity of using small time steps means that many steps must be calculated in order
to adequately sample conformational space. Therefore, molecular dynamics calculations are
normally performed with class 1 force fields. Molecular dynamics simulations are analyzed by
following the trajectory of a variable with time, often angles for oligosaccharides. Most
modeling programs now can perform molecular dynamics calculations [347,462,466,489].
Quantum mechanical calculations [490,491,492,493] are now fast and accurate enough that
they have become the method of choice for studying conformations of individual monosaccharides and disaccharides [231,494,495,496,497]. The use of density functional theory [498]
allows the study of much larger systems. High-level methods have to be employed to yield
accurate results in terms of energies [499]. The continuing improvements in computational
power will result in acceleration of the use of these methods.
Acknowledgement
I would like to thank Al French for comments on this chapter from the previous edition.