Warwick, June 1-5, 2009
Multiscale modeling of
Warwick, June 1-5, 2009
Multiscale modeling of
lipid and surfactant systems
lipid and surfactant systems
Mikko Karttunen
Dept. of Applied Maths, University of Western Ontario
Web: www.softsimu.org. Email: mkarttu@uwo.ca
Acknowledgements
Younger
Papers, parameters & movies: www. softsimu.org
Slightly older
Teemu Murtola
PhD student
Andrea Catte. PDF
HDL cholesterol, nanodisks
with proteins
Maria Sammalkorpi (Princeton)
Tomasz Róg (Tampere/Helsinki)
postdoc, Academy of Finland Fellow
Ilpo Vattulainen (Tampere/Helsinki)
Mikko Haataja (Princeton)
postdoc
Samuli Ollila (PhD student)
Perttu Niemelä (at VTT)
Emma Falck (at Boston Consulting) Michael Patra (at Zeiss)
Funding & resources:
www.sharcnet.ca
...but where is London, Ontario...
To be more precise:
Dates back to 1793 - the city was founded 1826
Population: 352,395 (greater area: 457,720)
Downtown London.
What’s on the plate?
Movies: www.softsimu.org
Membrane dynamics: diffusion in single-component lipid
membranes. Structure has been studied a lot but dynamics has received
surprisingly little attention. A new mechanism is suggested.
Surfactants. Micellation & micelle fission
Coarse graining using structural data from MD. First using a simple
system of NaCl & water and keeping solvent, then moving to a solvent free
membrane.
The “Mercedes-Benz” model for water & cold denaturation.
Many time & length scales
How to bridge the scales - no single method is applicable in all cases.
Macroscale:
Simulation:
• times > 1 sec
• lengths > 1µ
• phase field models, FEM,…
Experiment:
Naked eye
speed
Light
microscope
200 nm
0.2 nm
ATOMS
• electronic structure
• ab initio, Green functions
Electron
microscope
MOLECULES
20 nm
2 nm
Subatomistic scale:
ORGANELLES
2 µm
accuracy
• times ~ 10–15 – 10-9 sec
• lengths ~ 1-10 Å
• Molecular Dynamics,
Monte Carlo
CELLS
• times ~ 10–8 – 10-2 sec
• lengths ~ 10-1000 Å
• DPD, coarse-graining
Entity:
0.2 mm
20 µm
Mesoscale:
Atomistic scale:
Scale:
Aim: multiple scales in time and space
Multiscale modeling of emergent materials: biological and soft matter.
Murtola, Bunker, Vattulainen, Deserno, Karttunen, Phys. Chem. Chem. Phys., 2009, 11, 1869
∂
∂ The state
dynamic variables
from the!
description.
of∂the system
c
ct) at this mesos
P (x,∂t)c==− −∇JVi · with
+ FCC
·
P
(x,
i
Jthat
= − ∇µ
∂xti=1:
c {Q
=
with
J+
=toJ̃,−
t Level
of description
byColloidal
,∂Q
PiSuspension
}.
the m
∂t is given
∂P
i−∇J
3.2
Hydrodynamics
3 Mesoscopic
Example:
A
i The FPE
i corresponds
ζ
ζ
#
"
level 2 is given by [22]
! ∂
∂
Pi
P (x,
t).
+
k
T
·ζ
(Q)·
+
If the
we
looksection
at
theJwe
motion
of" ∂P
the solvent
molecules,
we
will
see
# concepts
B a systematic
ij contribution
In mass
this
will
illustrate
the
fundamental
of
thet
where
flux
has
proportional
!
∂P
M
k
T
i ∂
j
i
B
∂mass flux J ij
∂
where
the
has
a
systematic
contribution
pr
CC
each
other
resulting
in
a
rapid
motion.
However,
if
we
look
“fro
V
·
P
(x,
t)
=
−
+
F
·
P
(x,
t)
different levels
of description
that
to J̃
describe
collo
i
i are used
tential ofseveral
the colloidal
particles
plus
ai stochastic
part
with
a
var
∂t
∂Q
∂P
i
tential
of=the
colloidal
particles
plus
aoftostochastic
part J̃
CCis imade
the
multitude
of
molecules,
a
collective
motion
will
be
appreciate
colloidal
suspension
of
a
collection
small
solid
objects
#
"
Here,
V
P
/M
,
F
is
the
effective
force
due
the
rest
of
colloidal
to the transport
coefficient
c/ζ.
the sake of simplicity, we hav
i
i
i
i
!For
∂
∂which
Pfluid
i thesimplic
particles)
of space
the
size
of,
say,
a
micron
suspended
in
a
such
as,
to
the
transport
coefficient
c/ζ.
For
the
sake
of
in
a
region
of
move
coherently
(overwhelming
small
e
exerted
on
particle
i
and
ζ
(Q)
is
a
friction
tensor
depends
on
the
po
P
(x,
t).
+
k
T
·ζ
(Q)·
+
ij
B a way that
suspension
is
dilute,
in
such
hydrodynamic
interactions
ij
Fick
Thermodynamics
∂P
∂P
iappreciate
j canM
i kbest
B T appreciat
A
roadmap
of
this
section
is
shown
in
Fig.
1.
the
colloidal
particles.
The
physical
picture
behind
(4)
be
collision).
It
will
be
possible
to
slowly
evolving
wav
ij
suspension
is dilute,
in such
a way
that hydrodynamic
Continuity equation
No
equation
ofone
motion
Otherwise,
obtains
non-local
in
space
equations
[7].
When
the
s
mathematically
equivalent
stochastic
differential
equations
(SDE)
sort of collective
motion.
The
variablesinthat
capture
these collectiv
CC
one
obtains
non-local
space
equations
we Otherwise,
may
use
the
ideal
gas
expression
for
the
chemical
potential
µ[7].
=
Here, Vi = Pi /Mi , Fi is the effective force due
to the rest of colloid
!
drodynamic
variables.
TheseClassical
variables
the mass
density
field
CC
3.1
Microscopic
Level:
Mechanics
2are
ζ
(Q)·V
dt
+
F̃
dQ
=
V
dt
and
dP
=
F
dt
−
j
iparticle
i iideal
i friction
to the
usual
diffusion
equation
D∇
c−
∇which
J̃ ijwhere
D
=
exerted
on
and ζ ijgas
(Q)∂texpression
isc a=
tensor
depends
ondk
the
i
we
may
use
the
for
the
chemical
po
Bi .T
density
fieldparticles.
gr (z), and
the
energy
density
field
defined
by
r (z),
j(4) e
the At
colloidal
The
physical
picture
behind
can
be
best
appreci
2
expression
for
the
diffusion
coefficient.
Equation
(6)
can
be
easily
the most
microscopic
level, we ∂
can
model
a colloidal
tomathematically
the usual
diffusion
equation
=equations
D∇
c(SDE)
− ∇suspensio
J̃ whe
t c!
equivalent stochastic differential
cretizing
thesolid
resulting
stochastic
diffusion
equation
with
We the
observe
that
the particles
evolve
according
to their
and
they
suspended
objects
are
spherical
and velocities
we
needfinite
onlythat
6diffe
deg
expression
for
the
diffusion
coefficient.
Equation
(6)
c
mδ(r
−
q
),
ρ
(z)
=
i their positio
r particles that!
jected
to
forces
due
to
the
other
colloidal
depend
on
describing
thedtstate
of
the object,
the
position
Qζi and
the moment
CC
discretization
technique
for
stochastic
partial
differential
equations
[
(Q)·V
dt
+
d
F̃
dQ
=
V
and
dP
=
F
dt
−
j
i
i
i
i
ij
i
cretizing
theFor
resulting
equation
with
and of
velocities,
−ζ
.stochastic
Note that
a diffusion
colloidal
jconsider
is moving,
ij (Q) · Vjobjects
mass.
irregular
we ifwould
need particle
also
to
oriei
!
j
Fokker-Planck
Smoluchowski
ert forces
on etc.
the technique
colloidal
particle
i.gstochastic
These
are
the result
of theare
hydro
(z) =forces
ppartial
q
discretization
for
differentia
locities,
The
fluid
in
which
solid
colloidal
su
rthese
i δ(r −particles
i ),
Friction tensor
Diffusion
that
are
captured
at this according
level of description
throughand
thethat
frictio
3.6 interactions
Macroscopic
Level:
Thermodynamics
We
observe at
that
particles
evolve
to
velocities
th
i their
scribed
thethe
most
microscopic
level by the
positions
qi and
mome
ζjected
the particles
arecolloidal
also subject
to
stochastic
forces
dtheir
F̃i that
ar
!
ij (Q).toFinally,
forces
due
to the other
particles
that depend
on
posit
of
mass
of
the
molecules
constituting
the
fluid.
Again,
we
assume
ethat
=
ei scales
δ(r
− qiniof),
matically
described
in(Q)
terms
of. Note
Wiener
processes.
The
variance
these
force
r (z)
Finally,
might
be
interested
in
very
long
time
which
the
andwe
velocities,
−ζ
·
V
if
a
colloidal
particle
j
is
moving,
j
ij
for
simplicity.
The
microscopic
state
will
be
denoted
by
z
=
{q
3.6
Macroscopic
Level:
Thermodynamics
by
the
Fluctuation-Dissipation
theorem
which,
at
this
level
of
description,
i
ert forces
onInthe
colloidal
particle
i.
These forces
are the are
result
the hyd
at equilibrium.
this
case,
the
only
relevant
variables
theofdynam
evolution
of
the
microstate
is
governed
by
Hamilton’s
equations,
form
dF̃i dF̃jthat
= 2k
ζ ij (Q)dt.at this level of description through the fric
B Tcaptured
interactions
are
those
coarse-grained
variables
that
are
independent
ofastime
due
towhen
pari
Finally,
we
might
be
interested
in
very
long
time
scales
If
the
colloidal
particles
are
very
far
from
each
other,
it
happens
where
δ(r
−
q
)
is
a
coarse-grained
delta
function
(a
function
ζ ij (Q). Finally, the
subject to stochastic forces
dF̃i that
i particles are also
∂H(z)
∂H(z)
of the
microscopic
Hamiltonian,
like
total
energy,
or
those
coarse
pension is dilute, we may expect
that the mutual
influence
among
colloidal
pa
,
Q̇
=
,forc
q̇
=
atfinite
equilibrium.
In
this
case,
the
only
relevant
variables
ar
i
i
matically
described
in
terms
of
Wiener
processes.
The
variance
of
these
small region and normalized
to
unity,
see
Fig.
2).
In
the
a
∂p
∂P
negligible
and
that
the
friction
tensor
is
diagonal,
this
is
ζ
=
δ
1ζ,
where
ζ
i
i
ij Hamilto
like mass
and
volume,
thati (the
are theorem
constant
parameters
ijin the
by
the
Fluctuation-Dissipation
which,
at
this
level
of
description
the
energy
of
particle
sum
of
its
kinetic
energy
plus
the
those
coarse-grained
variables
that
are
independent
of
tim
the friction
coefficient. In this
case, the
SDE
equivalent
to
the
FPE
(4)half
decou
∂H(z)
∂H(z)
volume
ofdthe
colloidal
suspension
be
Classical Mechanics
Hydrodynamics
form
F̃
F̃j = 2kBwith
Tof
ζ ijthe
(Q)dt.
i dcontainer
ṗcalled
= −Langevin
, equations.
Ṗcan
−asunderstoo
,
to
the
interaction
its
It
may
appear
a contrad
ineighbours).
i =Langevin
set
of
independent
equations,
The
equati
of
the
microscopic
Hamiltonian,
like
total
energy,
or
∂qifrom
∂Q
Collisions in ps
Collective
motions
If the
colloidal in
particles
are very far
each other,there
as it happens
wh
i equ
a confining
potential
the
Hamiltonian.
Obviously,
is
no
evels of description in a colloidal suspension.
Arrows
denote
the
direction
of
dilute
suspension
predict
that
the
velocity
autocorrelation
function
of
a
colloida
through
adilute,
set ofwefield
variables
(which
have,
in principle
an infin
pension
is
may
expect
that
the
mutual
influence
among
colloidal
mass
and
volume,
that
are
constant
parameters
inthet
thislike
level
ofexponentially.
description
because
we
are
interested
in
theexperiments
long time,
m the Classical Mechanics level at the lower
left
hand
corner
to
Thermodydecays
As
wewe
have
seen,
this
isthat
at variance
with
of
freedom).
However,
should
note
the
above
fields
involv
negligible
and
that
theMethods
friction tensor
diagonal,
this
is
ζ ij Karttunen,
=for
δijthis
1ζ,discrep
where
Español
Mechanics of Coarse-Graining’
in Novel
inbuoyant
SoftisMatter
Simulations,
clear
long-time
tail
for
neutrally
particles.
The
reason
topP.right
hand‘Statistical
corner
of
the container
offields
the
colloidal
suspension
can
bo
the volume
system.
delta
functions
are “smooth”
(which
have
atosmall
number
the
friction
coefficient.
In
this
case,
the
SDE
equivalent
the
FPE
(4)
deco
Vattulainen and Lukkarinen (Eds.), Springer
Verlag
(2004).
tween theory and experiments for neutrally buoyant particles can only be attr
•
&
backbone is a similar factor. For both maps
surfactants
important factors in dividing the map3 int
1.2. Lipids
Lipids and&Lipid
Bilayers
orientation of the glycerol plane. A minim
A
B
45
C
D
50
E
F
55
Figs. T. Murtola, PhD thesis
Figure 1.3: Examples of phases formed
by lipids in water solution. Polar headgroups are shown in red, hydrophobic
tails in blue, and water is not shown.
(A) A spherical micelle.
(B) A cylindrical micelle.
(C) A bilayer.
(D) An inverted hexagonal phase.
Bilayers can bend to form, e.g., vesicles
(E), and bicubic phases (F) are also posSchematic description of how SOM dat
sible.
Fig. 6
constructing coarse-grained representations (see
Adapted from ref. 114.
Membrane dynamics is vital!
Zimmerberg et al, Science, 2005.
Membrane dynamics
Falck et al., JACS 130, 44 -08; PRL 97, 238102 -06
Falck et al., BJ 87, 1076 -04; BJ 89, 745 -05
A challenge for simulations at many different scales.
Why?
The role of fluctuations in membranes has been not been studied yet even
the main transitions depends on them: continuous or weakly first order?
Mechanisms behind lipid diffusion are still not well understood
Living systems are not static (& they are typically out of equilibrium)
The Singer-Nicholson fluid mosic model is not enough to describe
dynamics
Biology: rafts, signalling, lateral pressure, interactions with proteins,
pore formation, etc.
In addition: lipid composition matters and in all eukaryotic membranes
cholesterol has a special role.
We start by looking at diffusion.
FIGURES
FIGURES
Membranes:
role
of
rafts
A Model 1
Niemelä et al., PLoS Comput. Biol. 3, e34 (2007)
Vaino et al., J. Biol. Chem. 281, 348 (2006)
a Activation in a raft
b Altered partitioning
Extracellular
Dimerization
Dimerization
Antibodies,
ligands
Signalling
An
lig
Signalling
B Model 2
Simons,Clustering
Toomre, Nature of
Reviews
Molec.
Cell Biol. 2000;
rafts
triggers
signalling
Munro, Cell 2000
FIG. 1:
GPI
GPI
Classic view: membranes are quite static. WRONG: Bilayers/membranes are dynamic!
FIG. 1: signalling, etc.
Biological systems are inherently complex at all levels; structure-function,
FIGURES
Membranes:
role
of
rafts
A Model 1
Niemelä et al., PLoS Comput. Biol. 3, e34 (2007)
Vaino et al., J. Biol. Chem. 281, 348 (2006)
a Activation in a raft
b Altered partitioning
Extracellular
Dimerization
Dimerization
Antibodies,
ligands
Signalling
An
lig
Signalling
B Model 2
Simons,Clustering
Toomre, Nature of
Reviews
Molec.
Cell Biol. 2000;
rafts
triggers
signalling
Munro, Cell 2000
GPI
FIG.GPI
4:
Classic view: membranes are quite static. WRONG: Bilayers/membranes are dynamic!
FIG. 1: signalling, etc.
Biological systems are inherently complex at all levels; structure-function,
the lateral pressure profile to alter the shape of the membrane
Effect
on
proteins
cavity occupied by the protein as it changes conformation
from the closed to an open state. Then the work ∆W can
The work against lateral pressure (p(z)) profile to change the shape of a cavity
be occupied
written by
as:a protein as it changes conformation from closed to open:
!
∆W = p(z)∆A(z)dz,
(1)
In the case of MscL, the difference between the non-raft and raft cases
where
is the change in the cross-sectional area of
is 3-9 k∆A(z)
BT. This strongly supports the idea that the lipid environment regulates the
This and
has also
strong
influence
on binding
affinities
and partitioning
theactivity.
protein
p(z)
is the
pressure
profile.
Here,
we use an
(cytochrome).
approach
identical to that used in ref. [62], and identical val-
ues
for ∆A(z) for MscL as used in ref. [62], in which the
These findings also provide support to the idea that changes in lateral pressure
area
unchanged
in the
middle
the-98).
membrane bemay is
be kept
very important
in general
anesthesia
(R. of
Cantor
tween the two states. Error bars for ∆W have been calculated
using results for different monolayers. It is, however, important to realize that ∆W depends on the second moment of the
lateral pressure profile [62] and thus is susceptible to small
changes of lateral pressure far from bilayer center. Therefore extra caution must be followed when interpreting these
More: Niemelä, Ollila, Róg, Vattulainen, Karttunen, J. Struct. Biol, (2007).
To jump or not to jump?
Falck et al., JACS 130, 44 -08; PRL 97, 238102 -06
Over 300 ns, systems from 128 to 4608 lipids. T=323 K
Existing paradigm: Lipid diffusion is rattle-in-a-cage, punctuated by jumps.
Experimental results differ by 2 orders of magnitude. Interpretation:
QENS: fast motion (König et al, J. Phys. II -92; Tocanne et al, Prog. Lipid. R. -94)
FRAP: slow, random walk motion (Vaz & Almeida, BJ -91)
rattle-in-a-cage has been demonstrated (Wohlert & Edholm, JCP, 2006)
random walk has been demonstrated (Sonnleitner et al. BJ 1999)
jumps have never been shown to exist - a hypothesis to interpret QENS exps.
Our goal: Study the physical mechanism(s) behind lipid diffusion.
For jumps to dominate: in a 30 ns trajectory one should observe about 4
discontinuous jumps per lipid. One can make a simple estimate using
!2 ∼ 4Dt with D ≈ 1.5 × 10−7 cm2 /s and ! = 0.7nm
In large systems, one should see 1000’s of jumps.
Short times: correlations
Falck et al., JACS 130, 44 (2008).
Over 300 ns, systems from 128 to 4608 lipids. T=323 K
Observation: in over 300 ns, less than 10 such jumps were seen (100 ps time scale)
Lipid diffusion cannot be
dominated by jumps
Diffusion of individual lipids over 30 ns
Then, what is the mechanism?
RED:
How do the lipids move in
relation to their neighbors?
Are the motions correlated?
If so, what is the range and
time scale?
Conclusion: in short time scales, motions
are strongly correlated, jumps do not
dominate.
Question: How about longer time scales?
jumps look like this
Short times: correlations
Falck et al., JACS 130, 44 (2008).
Over 300 ns, systems from 128 to 4608 lipids. T=323 K
Observation: in over 300 ns, less than 10 such jumps were seen (100 ps time scale)
Lipid diffusion cannot be
dominated by jumps
Then, what is the mechanism?
How do the lipids move in
relation to their neighbors?
Are the motions correlated?
If so, what is the range and
time scale?
Motions of nearest
neighbors over 1 ns.
Neighbor motions are
correlated, no
jumping out of cages.
Conclusion: in short time scales, motions
are strongly correlated, jumps do not
dominate.
Question: How about longer time scales?
Long times: collective motion
Falck et al., JACS 130, 44 -08; PRL 97, 238102 -06
Movies: www. softsimu.org
Let’s vary the time window (1152 lipids, about 20 x 20 nm):
50 ps interval
500 ps interval
5 ns interval
30 ns interval
Long times: collective motion
Falck et al., JACS 130, 44 -08; PRL 97, 238102 -06
Movies: www. softsimu.org
Flow patterns are not coupled
to fluctuations of any particular
structural quantity.
The concerted motions probably
arise from a complex interplay
between density fluctuations,
undulations and thickness
fluctuations, lipid interactions,
interactions between lipids
and solvent molecules.
These flow patterns may have an
effect on biological functions,
including signalling and pore
formation.
New paradigm: Lipid diffusion may be dominated by correlations and collective motions.
s
s
Let’s move from the flatlands
(c) 0.6 ns
to spherical objects.
!"#
Figure 7
(f) 2.0 ns
the same SDS molecules highlighted in
dual molecule. The behavior of the blue
Fission/fusion pathway
Observed micelle
fission pathway
Elongated
micelle
Interdigitating
stalk
Proposed bilayer
budding / fusion
pathway
Sammalkorpi, Karttunen, Haataja:
Model: J. Phys. Chem. B 111:11722 (2007)
Fission: JACS 130:17977 (2008)
Salt: J. Phys. Chem B. 113:5863 (2009)
Interdigitating
stalk
Micelles: fusion and fission
Why?
Membrane fusion and fission are fundamental to cellular function and survival.
Examples: endo- and excytosis, recycling, viral entry & drug delivery
All of the above are inherently dynamic processes involving complex kinetics
Fusion - lot of research has been done (Jahn & Grubmüller):
X-rays: evidence of a short stalk (Yang & Huang)
Simulations: pore mediated pathway (Marrink & Mark)
Fission:
Difficult to access experimentally: pioneering work by Rharbi & Winnick who
showed the importance of electrostatics on fragmentation
Computationally: Pool and Bolhuis were the first to simulate fission with solvent and
to study transition paths. Markvoort et al.: existence of a short stalk using CG-MD.
SDS & Initial configuration
200-400 ns microscopic simulations: total over 2µs
Parameters available at www.softsimu.org
red: negative
Simulation engine: Gromacs
Random initial configuration
Explicit water: SPC
NpT ensemble
Force-field: Gromacs/Gromos
Explicit counterions & salt
SDS model: verification of
charge distribution with Gaussian
Constraints: LINCS (SDS),
SETTLE (water)
: of special importance Electrostatics: PME
A wide range of temperatures, and surfactant and salt concentrations
fate molecule with the employed Gromacs atom types
were studied.
Micellation: salt & temperature
CaCl2:323 K
Fully 3D. Periodic boundary conditions
NaCl; T=323 K
T=293 K; no salt
T=303 K; no salt
Size distribution & evolution
SDS molecules
Band: micelle
fusion events:
strips combine
fuzziness:
classification was
problematic
Transition: 288 - 297 K
test systems: 400 SDS & 200mMol with 50,000 waters
200
2
150
1
100
1
50
5
T=253 K
0
SDS molecules
0
(b) T = 293 K
SDS molecules
(a) T = 273 K
2
150
1
100
1
50
5
T=273 K
0
0
0
50 100 150 200
Time in ns
2
150
1
100
1
50
5
T=313 K; elongated T=323 K; slightly elongated
(d) T = 323 K
50 100 150 200
Time in ns
T=283 K
200
T=293 K
0
0
(c) T = 313 K
0
0
50 100 150 200
Time in ns
200
0
T=273 K; crystalline T=293 K; spherical
50 100 150 200
Time in ns
T=263 K
50 100 150 200
Time in ns
T=303 K
0
0
50 100 150 200
Time in ns
Animation of fission
Starting point: large micelle
from simulations with CaCl2
Procedure: Remove CaCl2
Provides access to
micelle fission kinetics:
size changes
surfactant motion
deformations
leakage
complexation with large
molecules
free energy changes
Sammalkorpi, MK, Haataja, JACS 2008
T=323 K, N(SDS)=186.
Snapshots
Sammalkorpi, MK, Haataja, submitted
T=323 K, N(SDS) = 186; pre-equilibrated for 200 ns
Decrease in salt
concentration:
Interdigitation: almost complete
After 4 ns: formation of a
dumbell with a long stalk
Diameter of the neck: only
slightly larger than the length of
an SDS molecule
High degree of ordering: the
molecules almost almost gel-like
After 6 ns: two micelles of (about) equal size
Areas of negative curvature and
splay-like conformations
High degree of ordering:
neighbors are highly correlated
Agreement with experiments:
increased salt -> decrease fission
rate (Rharbi & Winnick)
Animation of fission: halt
NOTE: Periodic boundary conditions
T=323 K, N(SDS)=186.
It is possible to control and
even to halt fission by varying the
salt concentration and/or temperature.
Intermediate maintained: for 30 ns
(previous: fission after 6 ns)
Stalk (transient) looks crystalline:
Ordering: neighbors are highly correlated
Interdigitation: almost complete
Diameter of the neck: length of an SDS
(a) T = 273 K
(b) T = 293 K
Physical mechanisms 1
Importance of electrostatic interactions:
Upon changing the ionic strength, the
Coulombic screening length changes
which leads to strong fluctuations.
Consequences:
Fluctuations lead to the formation of the dumbell which shape fluctuates
very strongly.
Formation of a highly intedigitated neck:, stretchable and stable; low
mobility, no contact with water.
Counterions have a dual role: In a dilute system, counterions are not
bound to the micelle but escape to the solution -> instability. But the same
counterions help to stabilize the stalk which is cylindrical (condensation)
Physical mechanisms 2
Lord Rayleigh, Phil. Mag. 14, 184 (1882).
Deserno, Eur. Phys. J. E 6, 163 (2001).
Rayleigh instability:
Surface tension wants to minimize the area but
the electrostatic repulsion leads to deformations.
When the size of the droplet increases, capillary
instabilities will break the droplet.
Difficulty: Charge neutrality (Deserno 2001); the micelle is charged
and surrounded by salt and counterions. Seen as pearl-necklace
conformations in polyelectrolytes (Micka, Holm, Kremer, 1999).
Ion condensation: Ions can condense on the surface or they can
even penetrate the micelle. The two lead to different scenarios
No penetration: condensation on the surface leads to screening of
the electric field - the droplet size is increases
Penetration: The Bjerrum length plays a crucial role and the
equilibrium droplet become very large
Cholesterol
• In membranes (eukaryotic cells)
• Four fused rings
• Precursors for steroid hormones
and bile acids
– Sex hormones
– Regulation of Na+
– Anti-inflammatory properties
– Vitamin A: vision and pigmentation
– Vitamin D: formation of bones
– Vitamin E: antioxidant
– RAFTS! Cholesterol seems to be
unique in its ability to enhance raft
formation!
Nelson & Cox: Lehninger Principles of Biochemistry, 3rd ed.
Cholesterol
• In membranes (eukaryotic cells)
• Four fused rings
• Precursors for steroid hormones
and bile acids
– Sex hormones
– Regulation of Na+
– Anti-inflammatory properties
– Vitamin A: vision and pigmentation
– Vitamin D: formation of bones
– Vitamin E: antioxidant
– RAFTS! Cholesterol seems to be
unique in its ability to enhance raft
formation!
Nelson & Cox: Lehninger Principles of Biochemistry, 3rd ed.
and Length Scales in Biological Systems
Phase diagram: DPPC + cholesterol
: Experimental
phase
diagram
of
DPPC/cholesterol
mixture
(data
M. R. Vist and J. H. Davis, Biochemistry 29, 451 (1990)
Lipids at different resolutions
Figs: T. Murtola PhD thesis
1.5. Modeling and Simulations
United-atom model:
aliphatic hydrogens
are not represented
explicitly
Coarser particle models:
chemical identity is lost.
Focus on generic behavior.
Semi-atomistic/superatom
model where each bead
describes a few heavy
atoms.
Continuum models: describe
a bilayer as an elastic manifold
(upper) and/or to describe local
structure (lower)
ost and focus is on generic behavior. (D) Co
Groups
Fig. 6 Schematic description of how SOM data could be used
DPPC-chol CG models
Figure 4.1: C
for details).
positions of a
layer. Chole
gle particle, a
DPPC. Paper
dom to the ta
All the models describe a single monolayer of the b
Hence,
membrane
undulations
and
interactions
be
[II] T. Murtola, E. Falck, M. Karttunen, and I. Vattulainen. 2007. Coarse-grained model for phospholipid/
cholesterol bilayer
employing inverse
Carloare
with thermodynamic
constraints.
JCP 126:075101.
although
theMonte
latter
implicitly
included
in the eff
[III] T. Murtola, M. Karttunen,
andlateral
I. Vattulainen.
2009. Systematic
coarse-graining
from structure describe
using
that
the
structure
can
be
adequately
internal states: Application to phospholipid/cholesterol bilayer. JCP, accepted.
interacting through isotropic pair interactions. The
(COM) positions of (parts of) the molecules. The e
[I] T. Murtola, E. Falck, M. Patra, M. Karttunen, and I. Vattulainen. 2004. Coarse-grained model for
phospholipid / cholesterol bilayer. J. Chem. Phys., 121:9156– 9165.
Henderson’s theorem
Volume 49A, number 3
PHYSICS LETTERS
9 September 1974
Text
A UNIQUENESS THEOREM FOR FLUID PAIR CORRELATION FUNCTIONS
R.L. HENDERSON
Department of Physics, University of Guelph, Guelph N1G 2W1 Ontario, Canada
Received 5 August 1974
It is shown that, for quantum and classical fluids with only pairwise interactions, and under given conditions of temperature and density, the pair potential v(r) which gives rise to a given radial distribution function g(r) is unique up to a
constant.
Attempts are often made to deduce information
function, the thermal average of the second term in (1)
on molecular interactions in the liquid state by analysis
is
Henderson
theorem function
is analogous
to the Hohenberg-Kohn theorem
of theThe
measured
radial distribution
g(r)
136, B864of(1964)):
[e.g. 1(Phys.
]. UnderRev.
the assumption
only pairwise inter(! ~ o(ri-ri)) =
r').
(2)
actions, this work proceeds, for instance, by solution
of one of the common integral equations or by fitting
In the
thermodynamic
limit, n number
2 becomes n 2N,
g(r), where
The
electron
density,
together
with
the
(known)
electron
the measured data with computer simulation results.
n is the average density, and g(r) is the radial distribucompletely
defines
thepotential
Hamiltonian
of
system (within an additive
It is usually
assumed that,
once a pair
o(r) is
tionthe
function.
foundconstant).
which reproduces a given g(r), it is the only one
The uniqueness theorem follows from an inequality
which will do so. This unique relation is manifest in
for the Helmholtz free energy. We consider two systhe integral equations, but, so far as this author is
tems under identical conditions of temperature,
aware, has not been demonstrated beyong the context
volume and density, and described respectively by
!f drdr'o(r-r')n2(r,
A
Henderson’s
theorem
will give rise to
a unique pair correlation function
(in the
canonical
ensemble).
Henderson’s
theorem
assertsby
that
A (classical
or quantum)
system
described
thethe
Hamilto-Given that
two the
pro
reverse
is also true:
Two
systems,bywhich
have a Hamiltonian
Classical
or a quantum
system
is described
the Hamiltonian
nian
a const
inequality
holds:
of the form (1) and which feature the same
have pair
It mu
potentials which differ at most by a trivial constant.
uniquen
(1)
This uniqueness theorem follows as a beautiful application
of the Gibbs-Bogoliubov inequality. For two systems with
where the trace
Hamiltonian
and
the
following
inequality
holds
for
It gives a unique pair correlation function g(r). Hendersonʼs theorem says that the g(r) gives a
The proof
willfree
give
rise
to a unique
pairbecorrelation
(in space.
the
their
energies:
unique Hamiltonian
up
to a constant.
That can
proven usingfunction
the Gibbs-Bogoliubov
inequality
equality
canonical ensemble). Henderson’s theorem
asserts
that
the
canonical average proper for H1Give
(2)
reverse is also true: Two systems, which have a Hamiltonian
inequal
of the form (1) and which feature the same
have pair
where
denotes the (canonical) average appropriate for
potentials
at most
a trivial
twoby
systems
withconstant.
Hamiltonians H1 and H2
. The key which
point isdiffer
that equality
holds
if and
only
if
uniqueness
theorem
as a beautiful
application
isThis
independent
of all
degreesfollows
of freedom,
which implies
of the
the
Gibbs-Bogoliubov
inequality.
For two
systems
The above
holds
if pair
and
only
if H2-H1can
is independent
all adegrees
of freedom.
g1(r) and g2(r)
that
potentials
differ
onlyofby
constant.
See
theThenwith
The
samewhere
inequat
can differAppendix
only
by a constant.
Hamiltonian
and
the
following
inequality
holds
for
for a proof of this inequality.
butions but
state
space.
Consider
now two systems which are identical in all retheir
free energies:
on
some equality
Hilbert
Donʼt believe
the
above?
Try
the
following:
assume
that
there
are
2
systems
that
are
identical
spects except that the pair potential in one is
and the pair
using the spectra
except that the pair potentials are different.
potential in the other is . The corresponding two parti(2)
the classical and
distributions are
and . The uniqueness theorem asThen, if gcle
1(r) and g2(r) are identical, the above says that
ity only holds
serts
that Write
if down
, then
a constant.
Now,
u2(r)-u1(r)=constant.
the
freethe
energies
for is
both
systems
youʼll if
end upfor
with 0<0.
where
denotes
(canonical)
averageand
appropriate
differ
by point
more than
justequality
a constant,
the same
holds
forif
. The
key
is that
holds
if and
only
, and thus equality in (2) cannot hold, i.e., we have
if
the classical case
A particularly
Inverse
Monte
Carlo
Inverse
Monte
Carlo scheme
DIRECT
PROPERTIES
MODEL
Radial distribution functions
Interaction potential
INVERSE
Inverse Monte Carlo:
Dissipative Particle Dynamics:
• Reconstruct potentials from experimental RDFs
• Construct potentials from detailed simulations
• Coarse-grained description
• Energy transfer to microscopic degrees of
freedom via collisions
• Produces the canonical ensemble
Fi =
!
FijC + FijD + FijR
j !i
RDF
dissipative
conservative
random
1.0
V PMF ( r ) " # k B T ln g ( r )
Lyubartsev, Karttunen, Vattulainen, Laaksonen, Soft Materials -04
Murtola, Falck, Karttunen, Vattulainen, JCP -05, -07
IMC
Central idea of Inverse Monte Carlo: Adjust effective interactions to match the target RDF in
an iterative fashion.
Potentials: represented by a piecewise constant grid approximation
Vα
Sα
potential in bin
α
number of particle pairs in that bin
Relation to RDF:
Np
V
!Sα " = gα Np Aα /V
number of particle pairs in the system
total volume of the system
During each iteration, the derivatives of !Sα " with respect to Vβ can be calculated
for all pairs We can then express the changes in !Sα " to the first order in terms of
changes in Vβ as
∆!S" = A∆V
Aαβ
∂Sα
"Sα Sβ # − "Sα #"Sβ #
=
=−
∂Vβ
kB T
How
to
improve?
converged within the numerical accuracy of the Monte Carlo simulations, the number of steps
beTo
increased
to refine
the effective
interactions further.
minimize
finite-size
effects:
Surface tension constraint. To minimize finite-size effects to the effective interactions,
the simulations during the IMC procedure should be carried out with a
simulations during the IMC procedure should be carried out with a system that is identical in siz
system that is identical in size to the system from which the target
45
the system
from
which
the
target
RDFs
were
determined.
In some potentials
cases, the effective poten
RDFs were determined. In some cases, the effective
produced
thisdoway
do not generalize
to larger
produced
in thisinway
not generalize
to larger systems.
Forsystems.
example, in the present study
effective interactions for the largest cholesterol concentrations are too attractive to mainta
For example: effective interactions may become too attractive to
reasonably
uniform
density
in the system. Instead, larger systems form dense clusters separ
maintain
uniform
density.
by empty space. Such behavior is clearly unphysical, and may result from the absence of exp
However: larger systems form dense clusters separated by empty
effects of water in the model.
space, which is typically unphysical.
Such condensation effects can be characterized by the surface tension of the coarse-gra
One possible
surface
tension
model.
We define solution:
the surfaceuse
tension
γ of the
coarse-grained model as
!
"
$%
1 #
1
!Ekin " +
fij rij
,
γ=
V
2 i<j
Murtola,
Vattulainen,
(2007).
where
!Ekin " Falck,
= N kBKarttunen,
T is the average
kinetic JCP
energy
in the system, and the latter term is the vi
Surface tension
Condensation effects can be characterized by surface tension.
We define the surface tension γ of the coarse-grained model as

#
1 $
1 
γ=
!Ekin " +
V
2
i<j
%
fij rij 
virial
If this is close to zero or negative in simulations of small systems, larger systems may not be
stable. This is the case for the highest cholesterol concentrations.
Situations where thermodynamic properties, particularly the pressure, of the coarse-grained
model do not match the underlying atomistic model have also been encountered in other
coarse-graining approaches. Proposed solutions include:
1. iterative adjustment of the pressure followed by re-optimization of the interactions
2. imposing additional constraints on the instantaneous virial due to effective interactions
S. Izvekov and G. A. Voth, J. Chem. Phys. 123, 134105 (2005).
D. Reith, M. Putz, and F. Müller-Plathe, J. Comp. Chem. 24, 1624 (2003).
Notice
One should note that the surface tension cannot be directly related to the surface
tension in the atomistic simulations.
This is because the effective potentials are in general volume dependent.
Hence, the correct value of γ is not necessarily the same as the surface tension in the
atomistic simulations, which has been proposed to be zero in equilibrium.
Because of these considerations, the value of γ has to be fixed using other quantities.
Here, we used area compressibility
Models: note cholesterol alwats a single particle
Model I
Constrained in 2D
Speedup: 8 orders
of magnitude.
System size:
100 nm x 100 nm
Model II
Model III
- sn-1 & sn-2 are different
- bonded interactions
- centre of mass used
- internal state:
1. orderd
2. disordered
- Needs extra internal energy
terms
- sn-1 & sn-2 are different
- bonded interactions
- centre of mass used
- 7 non-bonded
- 10 bonded interactions
Coarse-grained
lipidmore
model carerivatives
requires
scope of the present discuss not close, there is no guaron of the change is an ascent
be too long.
namic constraints As de-
- Nielsen et al., PRE 59:5790 -99
of disordered tails (denoted as nd ) is required (or e
no , the number of ordered tails). The Hamiltonia
comes
"
H=
Sα Vα + ∆End + Efluct δnd2 ,
α
interactions,
in
particular
in
higher
cholesterol
concentrations.
Further,
severa
Model I
rnal states, i.e.,
al guess
for
fur10
DPPC-DPPC
out a constraint,
rent 5implemene very different
0
several of these
chol-chol
V(r) [kBT]
DPPC-chol
0.0
0.5
1.0
1.5
r [nm]
2.0
0.0
0.5
1.0
1.5
40r [nm]
2.0
0.0
0.5
1.0
1.5
2.0
2.5
r [nm] Modeling
Figure 4.2: Effective pairwise interactions from [I]. Adapted from [I]
Figure 4.1:
0%
for details)
4.7 %
positions o
12.5 %
layer. Cho
20.3 %
gle particle
DPPC. Pap
29.7 %
dom to the
50.0 %
Model II
V(r) [kBT]
10
Text
Text
head - head
tail - tail
tail - chol
head - sn-1
head - sn-2
head - chol
chol - chol
5
0
3
3
2
2
1
1
0
0
15
sn-1 - sn-2
40 bond
head - sn-2 bond
head - sn-1 bond
15
10
10
5
5
0
0
-5
-5
0.0
0.5
1.0
1.5
r [nm]
2.0
0.0
0.5
1.0
1.5
r [nm]
2.0
0.0
0.5
1.0
1.5
r [nm]
sn-1 and sn-2 are different
2.0
V(r) [kBT]
4
2.5
V(r) [kBT]
V(r) [kBT]
V(r) [kBT]
4
0%
4.7%
12.5%
20.3%
29.7%
4.4. Simulations with Coarse-Grained Models
V(r) [kBT]
Model III
5
43
Murtola, Karttunen, Vattulainen, accepted for publication in JCP
head - head
o-o
o-d
head - o
head - d
head - chol
d-d
0
0.02
E [kBT]
2
2
1
0
0.01
-1
-2
0
0
10
20
30
Efluct [kBT]
V(r) [kBT]
4
0.00
V(r) [kBT]
Chol. conc. [%]
o - chol
5
d - chol
chol - chol
40
0%
4.7%
12.5%
20.3%
29.7%
0
0.0
0.5
1.0
1.5
r [nm]
2.0
0.0
0.5
1.0
1.5
r [nm]
2.0
0.0
0.5
1.0
1.5
2.0
2.5
r [nm]
Figure 4.4: Effective pairwise interactions from [III]. Ordered and disordered tails
are marked with o and d, respectively. The small figure shows the energy difference
Cholesterol & clustering at 20%
0.0
Cholesterol at approx. 20 %
Text
1.0
80
60
y [nm]
0.8
S(k)
◦
40
0.6
0.4
20
0
0
0.2
Murtola, Falck, Karttunen, Vattulainen, JCP (2007).
20
40
x [nm]
60
80
4.5. Atomistic Simulations
Transferability between concentrations
1.5
0.4
1.0
0.2
5% 13%
S(k)
S(k)
S(k)
S(k)
Figure 4.6: Potential transferability in [
1.5
top diagram summarizes the transferab
0.4 5% 13%
0
1
2
tween
different
concentrations:
the first +
The first +/- stands for (qualitative) reproduction of
0.5
small behavior of S (k)
0.2
1.0
for (qualitative) reproduction of small k
The second + for qualitative reproduction of the
of S(k), the second + for qualitative rep
nearest-neighbor
0
1 peak2 in S (k)
0.0 nearest-neighbor peak in S(k), and
0.5
of the
0.2 20% 30%
sible
third + for quantitatively nearly cor
2.0
Third + for quantitatively nearly correct S (k)
away from the small k 0.1region. The figur
away
0.0 from the small k region
0.2 20% 30%
1.5show the transferability between two
tom
2.0
0.0 The solid lines
solid line: the correct0.1
S(k)
adjacent
concentrations.
1.0 0
1
2
3
1.5
dashed line: by the transferred interactions
correct S(k), and the dashed line the S(k)
0.0
1.0 0
1
2
3
the0.5
transferred interactions. The color of
shows
0.5
0.0 the simulation concentration. The u
0also representative
5
10the transfer
15
ure
is
for
In practise: very different potentials lead to the same RDF
0.0
/nm]
k
[2
the
simpler
models
away
from the small
0
5
10
15
P. G. Bolhuis and A. A. Louis, Macromolecules 35, 1860 (2002).
Conclusions
Fluctuations play an important role in defining membrane properties.
A new paradigm for lipid diffusion is suggested. Neighborneighbor correlations and concerted motion may dominate.
Biological importance: rafts, signalling, lateral pressure
Coarse graining using structural data. IMC is possible approach.
It does not come without problems but can be used to reach
systems sizes over 100nm x 100nm
Micelles: new fission mechanism for charged micells.