Coarse grained simulations of
Lipid Bilayer Membranes
P. B. Sunil Kumar
Department of Physics
IIT Madras, Chennai 600036
[email protected]
Atomistic MD: time scales ~ 10 ns
length scales ~100 nm2
To study phenomena like fusion, fission, domain formation, elasticity etc,
atomistic models are not useful.
How to explore length and time scales much larger than that is achievable
through atomistic MD ?
Continuum models, that treat bilayer as a single elastic sheet are are unable to
answer many questions where self assembly and bilayer fluid structure are
important.
A course grained model, that treat a group of atoms as effective “interaction
centers”, will be useful to bridge this gap
Some general features
• Reduction in the number of degrees of freedom
• Use of short range interaction potentials - finite cut off for LJ and
electrostatic interactions
• Smooth potentials and energy to enable large integration time
steps.
Goetz-Lipowsky Model
Journal of Chemical Physics 108, 7397 (1998)
Spherical particles are linked together to make a surfactant molecule
Only three types of particles: (a) “water”, (b) “hydrophobic” and ( c) “hydrophilic”
Interaction between (b) and (c ) are soft repulsive with
All other particles interact with LJ of the form
All interactions are cut off at
.
To avoid discontinuities due to cut off all potentials are shifted by
The particles within a chain are connected
by harmonic springs
Stiffness of the chain is maintained by a three
body potential;
If there is a preferred angle for the bonds this potential
Modified to
Characteristic time scale
Characteristic energy scale
Here m is the mass of the “particles” ,
and
energy scales that enters the LJ interaction
are length and
Different models for “lipids”
ht4 - flexible chain with one
head and four tail particles
HT4- Semiflexible chain with
one head and four tail particle
H3(T4)2 - semilflexible double
chain with 3 head particles.
Comparing the length of these chains with that of an actual lipid molecule like
DOPC we get a length scale
to be of the order of .35 nm .
This length scale selection implies that each “particle” represent two or three
CH2 groups. The mass of the particle is thus between that of water and that of
three CH2 groups - 16 g/mol <m Na < 56 g/mol
Similarly one can choose the value of
to be that between two water
molecules or that of a CH2 group giving . 42 KJ/mol < Nav < 1.20 KJ/mol
With these the characteristic time scale in the simulation is tsc= 1.5 ps . The
actual time step used is however tsc/2000 ~ .75 fs
Starting from a random configuration of “lipids”, Monte Carlo simulations were
used to obtain a bilayer structure.
This structure was then equilibrated using MD simulations using a leap frog
integrator.
A semi-quantitative coarse grained model for membranes
S. J, Marrink et.al J. Phys. Chem B 108,750 ( 2004)
As before many molecules are clubbed together in to few interaction centers
The idea here was to parameterize the interactions such as to get a
realistic lipid to make quantitative predictions
Interactions are tuned by comparing with MD simulations for a variety of phases
and components
Model
Interaction sites are of four different
types
(P) polar - Hydrophilic groups
(N) non - polar - mixed polar and apolar
groups
( C) apolar - Hydrophobic groups
(Q) charged
N and Q can be of four types
(0) No- hydrogen bonding
(a) Acceptor
(b) Donor
(c) Both acceptor and donor
Interactions
All non-bonded particles
interact via
Five different values of
are used
(I) 5 KJ/mol (II) 4.2 KJ/mol (III) 3.4 KJ/mol (IV) 2.6 KJ/mol and (V) 1.8 KJ/mol
For electrostatic interactions
an
=20 was used
The effect of “hydration shell” was taken into account by using a smaller
effective charge for small ions.
In all cases of LJ interaction
nm
All interactions are cut off at
nm
As before a shift function is used to make the potential and forces go to zero
smoothly at
Bonded interactions are represented by linear springs with
spring constant kbond=1250 KJ/mol-nm2 allowing for 15%
variation in bond length
Bond angles are fixed by
with krad=25KJ/mol-rad2 allowing for 30% variation in angle
With these parameters the time scale per step is 30-40 fs
Example: Alkane +Water
4 H2O in one P
Link up C particles to form
Butane, Hexane etc.
4 CH2 in one C
Mass of the C particle is adjusted whenever the number of CH2 is not a multiple
of 4
The average C-C-C bond angle is kept at 140-142o
The “bonds” connecting “CM of 4 carbon atoms” in MD makes an angle of 136o
Temperature = 300 K.
Salt is represented by “hydrated ions” with 6 H2O surrounding an ion. The Mass
is calculated accordingly and an effective charge of .7 is used
Hydration implies hydrogen bonding , hence dissolved charges have an
additional “da” interaction.
MARTINI force field --
Marrink et. al J. Phys. Chem. B 2007, 111, 7812-7824
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Vesicle simulated with
periodic boundary
conditions was fused
with its periodic image
Starting from preformed
stalk, hemifused state
was formed within 2 ns
In one out of six
simulations, a transient
pore formed close to the
stalk, resulting in the
mixing of DPPC
lipids between the outer
and the inner leaflets.
PH induced fusion mechanisms of DPPC/palmitic acid at 1:2 ratio. T= 350–370 K
Solvent free models
Common feature: Attractive potential between lipids
It turns out that simple LJ pair wise attractive potential will not be sufficient
to get self-assembly
Brownian dynamics and Monte Carlo using hard core and soft core potentials
have been tried.
Advantage: Since all the time is spend in updating the membrane configuration
alone, larger length and time scales can be achieved .
Disadvantage:
• Cannot be used to study dynamics. Absence of solvent leads to wrong
conservation laws
•Mapping of interactions with real systems is hard.
•The model can be used to study broad physical features of the membrane at
equilibrium.
•Since internal degrees of freedom are included, effect of gel-fluid transition
can be studied
A modified LJ, in which the attractive basin is widened to make it longer
ranged, was used by Cook and Deserno: JCP 123, 224710 (2005)
The Lipids are represented by one head bead and two tail beads
All beads interact via a shifted LJ of the form
The bonds are represented by FENE springs
The lipids are straightened by an harmonic spring connecting the head and
the second tail particle.
an attractive interaction between the tail beads mimic hydrophobic interaction
induced attraction between lipids,
Cooke and Deserno introduces two different forms for the attractive
interaction Vcos( r ) and Vflat LJ ( r )
Solvent free models with soft core interaction
Joel D. Revalee, Mohamed Laradji and Sunil Kumar : JCP (2007)
lipid molecules are modeled as semi-flexible amphiphilic linear chains
composedof soft beads: one hydrophilic bead, mimicking the lipid head
group, and three hydrophobic particles, mimicking the lipid tail group.
BD using a Langevin thermostat with interaction potentials given by
soft pairwise interaction between neighboring particles , U(0) , is given by
The equations of motion,
The integration time step
value set by fluctuation dissipation.
are integrated using the velocityVerlet algorithm with
ensures average temperature to the
The value of the area per lipid of the molecule in the fluid phase
Comparing this with the exp. Value of al for DPPE bilayers we get
Diffusion coefficient is found to be
experimental values we get
, comparing with
The phase behavior of the lipid
bilayer can be characterized by the
diffusivity of the lipids, the chainorientational order parameter, the
bond-orientational order parameter,
and translational and bondorientational correlation functions
Solid circles correspond to a micellar phase, pluses
correspond to stable bilayers, and open circles
correspond to defective bilayers or an isotropic
solution of lipids
The lateral lipid diffusivity of lipids center of mass
The lipid sixfold bond-orientational order
parameter
The positional and bond-orientational correlation functions
kBT=2.70 and kBT=2.74