Structural and Thermodynamic Studies of Synuclein Proteins Related to Parkinson’s Disease: - Metallic Systems: A Quantum Chemist's Perspective - page 41

the.same.energy.is.directly.proportional.to.the.volume.of.the.phase.space,.i.e.,.the.
probability.of.each.accessible.microstate.is.equivalent.over.a.long.period.of.time..
Various. ensembles. from. canonical. to. grand. canonical. ensembles. can. be. used. in.
simulations. 47 .For.studies.of.biocomplexes,.the.canonical.and.isothermal-isobaric.
ensembles.are.usually.chosen..In.our.studies,.we.have.used.the.isothermal-isobaric.
ensemble.(NPT).in.which.the.number.of.particles,.the.pressure,.and.the.tempera-
ture.are.kept.constant..The.partition.function.can.be.described.using.the.canonical.
partition.function

∫ Z

Δ( N P T , , ) = ( N V T , , ) exp( − β PV C V ) d (2.1)

.

.

where
N .is.the.number.of.particles
P .is.the.pressure
V .is.the.volume.of.the.system
C .is.a.normalization.constant
β =

1 1 , . k B .is.the.Boltzmann.constant
T .is.the.temperature

− −

k T
B

The.Gibbs.free.energy.can.be.calculated

) .

(2.2)

G N P T ( , , ) = − k B InΔ( N P T , ,

.

In.order.to.more.accurately.mimic.the.conditions.of.experimental.studies,.the.tem-
perature. and. pressure. of. the. system. is. controlled. using. thermostats. and. barostats.
during. the. simulations.. Widely. used. thermostats. include. Berendsen. coupling, 48 .
Nosé-Hoover. temperature. coupling 49 . and. Langevin. dynamics. 50 . In. the. Berendsen.
thermostat,.the.system.is.weakly.coupled.to.an.external.heat.bath.whose.temperature.
T 0 .is.held.constant..The.temperature.of.the.system,. T ( t ),.is.adjusted.by.a.temperature.
coupling.time.constant.τ.according.to.the.formula:

dT t
dt

( ) T − T t ( )

0

. (2.3)

=

τ

.

The.Berendsen.thermostat.is.an.eficient.method.for.a.system.to.reach.a.desired.tem-
perature..The.Nosé-Hoover.thermostat.was.introduced.by.Nosé.and.developed.by.
Hoover..It.features.an.extra.variable. s .with.mass. Q .added.to.the.system..The.motion.
of.each.atom.with.mass. m i .and.velocity. v i .is.given.by.the.Nosé.equations

2

d r
dt

F
m s

2 v

s

ds
dt

i i i

= −

2

i

⎛
⎜

⎞
⎟

2

d s
dt

1

∑

2 2

= m s v − gk T . (2.4)

i i B 0

Qs

.

i