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Unformatted text preview: 18.417: Conformational Search via Molecular Dynamics Charles W. O’Donnell March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 1/35 Biological timescales for proteins Biological
process
occur
over
a
wide
range
of
3mescales,
for
example:
   Local
mo3ons
–
every
[10‐15
to
10‐1]
seconds:
   Hydrogen
bonds
vibrate
with
frequency
~10‐15s
   Sidechain
rota3ons
   Loop
fluctua3ons
 C C C H H H   Rigid
body
mo3ons
–
every
[10‐9
to
1]
seconds:
   Domain
movements
(e.g.
hinging)
   Enzyma3c
conforma3ons
changes
   Slow
–
every
[10‐7
to
104]
seconds:
   Helix/coil
transi3ons
   Dissocia3on
/
associa3on
   Folding
/
unfolding
 March 12th, 2009 calmodulin
 18.417 Lecture 11: Conformational Search via Molecular Dynamics 2/35 Molecular dynamics (MD) Molecular
dynamics
permit
the
study
of
complex,




























 dynamic
(i.e.
!me‐dependent)
processes
that
occur
in
biological
systems
 by
simula!ng
Newton’s
laws
of
mo3on.
 These
include:
   Protein
stability
   Conforma3onal
changes
   Molecule/molecule
interac3ons
   Ion
transport
 MD
concepts
can
also
be
used
in:
   Drug
Design
   Structure
determina3on
from
NMR
constraints
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 3/35 MD versus MC Molecular
Dynamics
(MD)
and
Monte‐Carlo
(MC)
simula3ons
o\en
 referred
to
in
the
same
breath
   Both:
   physically
model
protein
structure
   search
conforma3onal
space
by
small
itera3ve
steps
   MD
uses
Newton’s
laws
to
move
from
state
to
state
   Can
reveal
subtle
changes
   Search
depends
only
on
energy
(wanders
minima)

   MC
uses
arbitrary
steps
to
move
from
state
to
state
   Movements
not
based
on
physics
   Quicker,
can
exploit
extra
knowledge
to
improve
search
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 4/35 Physical protein modeling Every
“sequence

structure”
predic3on
problem
has
3
components…
 Molecular
Dynamics
no
different…
  REPRESENTATION
   what’s
a
protein…
atoms?
s3ck‐figures?
ideal‐spheres?
   what’s
the
protein’s
environment?
water,
etc?
  SEARCH

   check
different
protein
conforma3ons
through
in
silico
mo3on
  SCORE
   the
universe
wants
to
minimize
energy
and
maximize
entropy
   “best”
conforma3ons
those
with
minimal
energy
according
to
some
 
model
of
force
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 5/35 Representations Endless
diversity
in
representa!onal
choices:
 Full
atomic
resolu3on
vs.

 
ball‐and‐s3ck
polymers?
 All
hydrogens
vs.

 
only
polar
hydrogens?
 Explicit
water
molecules
vs.
 
implicit
solvent?
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 6/35 Motion How
to
determine
the
posi3on
of
an
atom,
given
its
last
known
posi3on?
 Newton’s
equa3ons:
 2 F = m⋅ a dU dr =m 2 ⇒− F = −∇U ( r) dr dt This
relates
the
deriva3ve
of
the
poten3al
energy
(U)
to
changes
in
 posi3on
(r)
as
a
func3on
of
3me.
 € By
simple
integra3on,
posi3on
is
defined
by
 2 































































or

 ˙ r = r0 + r0 t + ˙˙0 t r where:
 1 dU a=− ⋅ m dr € March 12th, 2009 r = r0 + v 0 t + at 2 18.417 Lecture 11: Conformational Search via Molecular Dynamics 7/35 Motion Thus
a
protein’s
mo3on
trajectory
can
be
determinis3cally
computed,
 given:

   The
ini3al
posi3on
of
all
atoms
   The
ini3al
distribu3on
of
veloci3es
of
all
atoms
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 8/35 Numerical integration r = r0 + v 0 t + at 2 Many
ways
exist
to
integrate
mo3on
(e.g.





























)
for
discrete
 values
(so
it
can
be
computed)
   For
example:


(Forward)
Euler
Method:
 € r ( t + δt ) − r ( t ) ˙ r( t ) ≈ δt r ( t + δt ) x(t) € δt r t r( t + δt ) = r( t ) + δt€r( t ) ⋅˙ € € €   Many
other
methods
exist
(Backward
Euler,
Runge‐Kuja,
etc)
   € Many
inapplicable
to
MD
because
of
imprecision,
complexity,
or
 non‐symplec3c
 € March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 9/35 Verlet algorithm r t ± δt Approximate





at












using
Taylor
series:
 € € then
sum
both
equa3ons:
 The
velocity
can
be
back‐computed
by:
 Advantages:
easy
to
implement,
low
storage
requirement
 Disadvantages:
prac3cal
precision
loss,
implicit
velocity,
ini3al
state?
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 10/35 Leapfrog algorithm Instead,
use
the
approxima3ons:
 “Leapfrog”
between
compu3ng































and

 Velocity
at
t computed
as:
 Advantages:
explicit
velocity,
doesn’t
take
difference
of
large
numbers
 Disadvantages:
posi3on
and
velocity
are
not
synchronized

 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 11/35 Velocity Verlet algorithm Use
the
approxima3ons:
 




















can
be
rearranged
as:
 Advantages:
posi3on
/
velocity
/
accelera3on
synchronized
















 























without
loss
of
precision
loss
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 12/35 Beeman’s algorithm Approxima3ons
yield:
 Advantages:
similar
to
Velocity
Verlet…
more
accurate
 Disadvantages:
computa3onally
expensive
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 13/35 Can MD fold a protein? Even
the
fastest
folding
proteins
fold
in












seconds…
 10−4 
Our
integrators
operate
with

 δt = 10−15 s € € http://www.research.ibm.com/journals/sj/402/allen.html Even
the
fastest
supercomputer
might
produce
on
the
order
of
 nanoseconds
(




















)
per
day!
 10−9 − 10−7 Typically
you
need
to
run
mul3ple
trajectories
to
remove
stochas3c
effects
 € March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 14/35 Multiple time-step (MTS) integrators Since
protein
interac3ons
occur
at
different
3mescales





















 (covalent
bond
vibra3on
vs.
domain
hinging)

evaluate
the
impact
of
 fast
and
slow
forces
separately:

 F fast + Fslow = m ⋅ a Evaluate
forces
using
train

 of
pulses
at
different
frequencies:
 € Advantages:
Orders
of
magnitude
faster
 Disadvantages:
Can
be
imprecise
 Izaguirre, et al, 1999 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 15/35 Long vs. short trajectories Open
Ques!on:

 
Do
long
trajectories
tell
us
anything
different
than
a
combina3on
 of
short
trajectories?

(only
one
way
to
find
out)
 BlueGene
project
‐
IBM
 
One
goal
was
to
use
a
single
supercomputer
to
run
very
long
 simula3ons
 Folding@Home
‐
Pande
group
(Stanford)


 
Use
distributed
compu3ng
(based
on
distributed.net,
 SETI@Home)
to
run
many
many
short,
parallel
MD
trajectories
–
 combine
trajectories
to
form
a
global
picture
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 16/35 Potential energy function Poten3al
energy
func3on
(some3mes
called
force
fields)
used
to
solve
for
 1 dU a=− ⋅ m dr Many
different
models
are
used
(AMBER,
CHARMM,
etc),
 
most
use
the
following
assump3on:
 €         Covalent
bonds
cannot
be
broken
 Implicit
treatment
of
quantum
mechanics
 Allow
for
experimentally‐derived
parameters
 Use
the
independent
assump3on:

 U = U bonded − atoms + U non − bonded − atoms March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 17/35 Covalently bonded terms Four
commonly
used
contribu3ons
for
covalent
bonds
 
(many
more,
e.g.
swope)
 U cov = U r bond +U θ bond − angle +U φ improper− dihedral +U ψ torsion Calcula3ng
poten3als
a
spa3ally
 local
opera3on
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 18/35 Bond term Model
the
force
of
a
covalent
bond
fluctua3ng
as
harmonic
spring
 U r bond = κ r ⋅ ( r − req ) 2 € source: swift.cmbi.ru.nl/teach/B1SEM March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 19/35 Bond-angle term Angle
formed
between
2
bonds
also
modeled
as
harmonic
spring
 U r bond − angle = κθ ⋅ (θ − θ eq ) 2 € March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 20/35 Improper dihedral angle term Angle
between
AB
bonds
and
CD
bonds
in
figure
below:
 source: hidra.iqfr.csic.es/man/dlpoly/USRMAN/ Also
modeled
as
harmonic
spring
poten3al:
 U φ improper− dihedral March 12th, 2009 = κ φ ⋅ (φ − φ eq ) 2 18.417 Lecture 11: Conformational Search via Molecular Dynamics 21/35 Torsion angle term Torsion
angle
is
a
period
term
reflec3ng
the
preference
for
the
electron
 clouds
of
like
charges
to
distance
themselves
 Modeled
periodically
with
cosine
term:

 “A Practical Introduction to the Simulation of Molecular Systems,” Martin Field 3 U ψ torsion νn = ∑ (1 + cos( nψ − γ )) n =1 2 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 22/35 Non-covalently bonded terms Two
major
forces
play
role
between
non‐bonded
atoms:
 Two
major
forces
play
role
between
non‐bonded
atoms
   Van
der
Waals’
force
   Quantum‐dynamic
dipole
forces

 
induced
when
electron
clouds

 
near
each
other
   Coulomb’s
Law
(electrosta3c
force)
 U non − cov = U electrostatic + U van − der−Waals € Calcula3ng
poten3als
a
spa3ally
 distance
opera3on!


 Computa!onally
complex!
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 23/35 Electrostatic term Standard
opposites‐ajract
kind
of
stuff…
 Strength
of
interac3on
dissipates
on
the
order
of
(1/distance)
 
Very
long‐range
interac3on
compara3vely
 U electrostatic qi q j = 4 πε0ε1 ⋅ rij € March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 24/35 van der Waals term van
der
Waals
force
quite
complicated
analy3cally
 
contains
both
an
ajrac3on
and
repulsion
force
 Very
strong
repulsion
 weak
aOrac!on
 strong
aOrac!on
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 25/35 Lennard-Jones potential Lennard‐Jones
poten3al
chosen
as
an
approxima3on
to
van
der
Waals
force
 U van − der−Waals € B C ij ij ≈ U Lennard − Jones = 4ε 12 − 6 r r ij ij 12 r 








term
only
an
 approxima3on
to
ease
 calcula3on

(( r 6 ) 2 ) € source: wikipedia March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 26/35 Lennard-Jones switch cutoff Since
Lennard‐Jones
is
s3ll
 computa3onally
complex…



































 Frequently
poten3al
is
ignored
past
 an
arbitrary
a
certain
Å
distance
 3
common
methods:
   Trunca3on
   Global
scaling
   Sigmoidal
switch
(best) 


 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 27/35 Missing / implicit energy terms? Hydrophobic
effect
   The
property
of
non‐polar
molecules
to
seclude
and
pack
against
 themselves
when
in
water
   Both
an
energe3c
and
entropic
force
   One
of
the
main
driving
forces
in
protein
folding!
   Not
accounted
for
in
these
poten3al
energy
calcula3ons…
implicit March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 


 28/35 Potential energy scale summary March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 29/35 Energy barriers Breadth
of
conforma3onal
search
depends
on
velocity
(i.e.
Temperature)
 Biologically‐relevant
temperatures
(23°C)
can
be
slow
to
sample
vastly
 different
conforma3on
because
of
energy
barriers
in
poten3al
func3on
 U ( s) s March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 30/35 Replica exchange One
common
technique
for
overcoming
energy
barriers
is
replica
exchange
 Conceptually:
   Run
mul3ple
simula3ons
concurrently,
but
at
different
temperatures
   Every
N
seconds
randomly
shuffle
conforma3ons
from
one
simula3on
to
 another
 K 600 
 380 
 273 
 200 
 replica
4 
 swap 
 swap 
 replica
3 
 replica
2 
 replica
1 
 time March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 31/35 Further: periodic boundaries Allows
you
to
simulate
more
realis3c
environment
with
limited
number
 of
molecules
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 32/35 Further: special solvent treatment Solva3on
shells
 March 12th, 2009 Ac3ve
site
solva3on
 18.417 Lecture 11: Conformational Search via Molecular Dynamics 33/35 Further: continuum electrostatics March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 34/35 Further: ensemble choice MD
simula3on
must
choose
a
set
of
sta3s3cal
mechanical
assump3ons
to
 operate
under
(if
it
wants
to
agree
with
thermodynamics)
   Canonical
ensemble
(NVT)
   Most
commonly
used
   Temperature
controlled
by
means
such
as
Langevin
dynamics
   Microcanonical
ensemble
(NVE)
   Less
common
   Isothermal‐Isobaric
ensemble
(NPT)
   Can
be
used
when
dealing
with
membranes/etc
 March 12th, 2009 18.417 Lecture 11: Conformational Search via Molecular Dynamics 35/35 ...
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This note was uploaded on 06/16/2011 for the course MATH 18.417 taught by Professor Jérômewaldispühl during the Spring '11 term at MIT.

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