Structural and Thermodynamic Studies of Synuclein Proteins Related to Parkinson’s Disease: - Metallic Systems: A Quantum Chemist's Perspective - page 42

Here,. g = N d.o.f .or. g = N d.o.f .+.1,. N d.o.f .denotes.the.number.of.degrees.of.freedom,. k B .
is.the.Boltzmann.constant,. T 0 .is.the.bath.temperature,. r .is.the.position,.and. F .is.the.
force..The.Nosé.equations.may.also.be.written.as

⎛ ⎞

d
dt

γ = − 1 g T

T t

0

− 1

⎜ ⎟

τ N ( )

⎝ ⎠

N H − d o f . .

1 ds

dt

γ =

s

Q

τ = . (2.5)

N H −

N k T

d o f . . B

.

where.τ N−H .is.an.effective.relaxation.time.that.can.be.estimated.during.the.simulation.
Similarly,. pressure. is. controlled. by. coupling. the. system. to. a. pressure. bath. and.
relaxing.the.system.to.the.desired.pressure. P 0 .with.the.Berendsen.barostat

dP t
dt

( ) P − P t ( )

0

= . (2.6)

τ

.

P

where.τ P .is.the.pressure.coupling.time.constant.
Another.temperature.and.pressure.control.method.is.Langevin.dynamics,.where.
additional.damping.and.random.forces.are.introduced.into.the.system..This.thermo-
stat.and.barostat.maintain.the.kinetic.energy.of.the.system.and.therefore.maintain.
the.system.temperature.(or.pressure).at.a.constant.value

m d r t
dt

2

( ) dr t

dt

( )

i i

. (2.7)

= F r t ( ( )) − γ m R t + ( )

i i i i i i

2

.

where
F .is.the.force.on.atom. i .with.mass. m i .and.position. r i ( t ).at.time. t
γ i m i .is.the.frictional.damping.applied.to.the.atom
R i .are.the.random.forces.applied.to.the.atom

Interactions. between. particles. can. be. described. using. force. ields. or. quantum.
chemical.models,.or.via.mixing.force.ields.and.quantum.chemical.models.in.a.single.
simulation..In.classical.MD.simulations,.a.force.ield.refers.to.the.set.of.parameters.
from.which.the.potential.energy.is.calculated.as.a.function.of.the.atomic.coordinates..
These.parameters.are.derived.from.its.to.experimental.measurements.and.quantum.
chemical.calculations..The.most.widely.used.force.ield.parameters.in.the.studies.of.
biocomplexes.via.classical.MD.simulations.are.either.“all.atom”.or.“united.atom”.
models..The.all.atom.model.provides.a.parameter.for.every.atom,.whereas.the.united.
atom.model.treats.groups.of.atoms,.such.as.the.methyl.group,.as.a.single.particle.to.
reduce.the.system.size.and.thus.the.computational.time.