Structural and Thermodynamic Studies of Synuclein Proteins Related to Parkinson’s Disease: - Metallic Systems: A Quantum Chemist's Perspective - page 44

energy.and.is.deined.as.a.periodic.cosine.function,.where. K .is.the.force.constant,. n .
describes.the.periodicity.of.the.torsion.potential,.ϕ.is.the.torsion.angle,.and.δ.is.the.
phase.shift.
Besides.these.terms,.the.Urey-Bradley.term,.which.is.used.to.describe.the.inter-
action. between. atoms. separated. by. two. bonds. (Equation. 2.9). and. the. improper.
dihedral.term,.which.is.used.to.maintain.planarity.and.chirality.(Equation.2.10).are.
added.to.the.potential.energy.function.(Equation.2.8).

1
2

∑

2

. (2.9)

E

U − =

K [ 1 − cos( n φ )]

φ

.

1 4 ,

1
2

∑

] .

2

(2.10)

E

improper =

K

ψ ψ ψ

[ −

0

.

The.van.der.Waals.interaction.term.is.represented.by.a.Lennard-Jones.6-12.poten-
tial.(the.fourth.term.in.Equation.2.8).that.has.two.components,.a.repulsive.potential.
represented.by.the. r −12 .term,.preventing.atoms.from.overlapping.at.short.distances,.
and.an.attractive.potential.at.long.distance,.represented.by.the. r −6 .term..The.value.
of.ε.controls.the.depth.of.the.potential.well,.σ.is.the.distance.at.which.the.potential.
goes. to. zero.and. r . is. the. internuclear. separation. of. the. two. atoms.. Coulomb's. law.
is.used.to.calculate.the.electrostatic.interactions.of.any.two.atoms.(the.ifth.term.in.
Equation. 2.8). with. partial. charges. of. q 1 . and. q 2 . separated. by. a. distance. r 12 ,. where.
ε 0 .is.the.dielectric.constant.of.the.medium..For.simulations.of.the.solvent,.we.use.
the.transferable.intermolecular.potential.3P.(TIP3P).model.for.water.that.includes.
a.three-site.rigid.water.molecule.with.partial.charges.and.a.Lennard-Jones-van.der.
Waals.potential.on.each.atom. 52
The. classical. MD. simulations. presented. in. this. chapter. use. cubic. box. periodic.
boundary. conditions.. When. an. atom. leaves. the. box. from. one. side,. another. atom.
enters.the.box.on.the.opposite.side.of.the.box.with.the.same.velocity. 50 .Under.these.
conditions,. angular. momentum. is. not. conserved. because. of. the. lack. of. rotational.
symmetry..However,.the.linear.momentum.is.conserved..In.our.simulations,.the.sys-
tems.have.neutral.net.charges..If.this.were.not.the.case,.the.charge.on.the.ininite.
periodic.system.would.be.ininite..To.achieve.neutral.charge,.we.add.sodium.counter-
ions.to.the.system..The.minimum.image.convention.requires.a.spherical.cutoff.with.a.
length.of.the.half.box.length.for.calculating.nonbonded.interactions. 50 .Additionally,.
the. periodic. boundary. conditions. should. be. applied. with. extreme. care. because. a.
small.box.size.relative.to.the.size.of.the.biocomplex.could.result.in.interactions.of.
the.atoms.with.their.own.images.in.neighboring.boxes..Periodic.boundary.conditions.
can.be.used.with.methods.such.as.Ewald.sum.method, 53 .which.treat.long-range.elec-
trostatic.interactions..This.method.deines.these.interactions.via.calculation.of.the.
short-range.part.of.the.potential.in.real.space.and.the.long-range.part.of.the.potential.
in.Fourier.space..It.is.assumed.that.the.short-range.term.is.easier.to.calculate.than.the.
long-range.term,.and.thus.the.short-range.calculation.is.used.as.the.method.to.treat.
long-range. interactions.. The. Fourier. summation. is. applied. to. the. ininite. periodic.