A presentation on the use of Dissipative Particle Dynamics to study biophysical processes related to the cell, polymers and nanoparticles.

1. Towards a Computational Cell
Julian C Shillcock MEMPHYS
Source: chemistrypictures.org

2. Structure of talk
What are the organizational and dynamic properties of membranes
at a molecular level?
How are molecules trafficked among the organelles of a cell?
To answer these questions, we could reconstitute model systems in vitro or
we can build mathematical and computational models.
• Lipids + Water + Proteins + Self-Assembly = Life
• Q. Whither DPD Simulations?
• Case Study 1: Vesicles and Fusion
• Case Study 2: Nanoparticles and Endocytosis
• Ans. Simulations that do what you tell them
• Conclusions
MEMPHYS 2

3. Evolution of (Bio-) simulations
Past
Assembly – random mixture or a few structures
(essentially a passive view of the system; we can prepare it but we
cannot subsequently interact with it)
Present
Response – equilibrium properties & perturbations
Future
Control – we want to interact with a system as it evolves, keep only
molecular details necessary to create structure on the scales of
interest, observe self-organization and emergent phenomena
(The Middle Way Laughlin et al., PNAS 97:32-37, 2000)
MEMPHYS 3

4. Why not do Molecular Dynamics?
• Atomistic Molecular Dynamics is accurate at atomic
length-scale (but less useful for macroscopic properties
such as shape fluctuations, rigidity,…)
• Complex force fields capture motion at short time-
scale (bond vibrations, but probably irrelevant for large
supramolecular aggregates)
Atoms are not the whole story; there are organizing principles
above the atomic length scale
Fusion event (0.32 µsec. ) with DPD ~200 cpu-hours
Fusion event using all-atom MD ~500 cpu-years
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5. Complex Fluids
“Simple” fluids are isotropic
“Complex” fluids have structure
arising out of the “shape” of their
constituent molecules, e.g., liquid
crystals or lipids with different
headgroups or tail lengths
Multiple length and time scales,
e.g., lipid bilayers have a
membrane thickness of ~ 4 nm
but form vesicles/cells with
diameters from 50 nm to 10 µm;
lipids diffuse in ~100 ns but cells
Source: chemistrypictures.org
divide in ~minutes.
Plasma membrane and transport vesicles are composed
of hundreds of types of lipid and protein molecules
MEMPHYS 5

6. Lipids
Lipid molecules are amphiphiles and surfactants
(surface-active agents)
- Water-loving headgroup (1)
- Water-hating hydrocarbon tails (2)
When placed in water, lipids aggregate into distinct forms: micelle, vesicle, etc.
Aggregation is driven by the hydrophobic effect: tendency of water to
sequester oily materials so as to maintain its H-bonding network.
Properties of the aggregates depend on physical characteristics of lipid
molecules, e.g., their “shape”, headgroup size, tail length, as well as their
chemical structure.
Source: Wikipedia
MEMPHYS 6

7. Amphiphiles
How do we represent amphiphiles in a simulation? Two aspects:
- Chemical nature: polar headgroups bound to oily tail(s)
- Molecular shape: large or small head/straight or kinked tails
Molecular structure leads to a preferred shape in amphiphilic aggregates
Inverted
Cone
Cylinder Cone
Source: chemistrypictures.org
MEMPHYS 7

8. Headgroup Size
Amphiphile
architecture
HC6
4
H2C6
3
H3C6
2
σr 02/kBT
1
0
0.6 0.7 0.8 0.9
-1
-2
-3
Apr/Nr02
-4
Amphiphile architecture modulates planar bilayer response to tension
MEMPHYS 8

9. Bilayer Self-assembly in Water
324 lipid molecules in (invisible) water
Hydrophilic headgroup
Hydrocarbon tails
Simulation Notes
Water is present in all movies, but invisible
to reveal dynamics of processes.
Periodic Boundary Conditions are used,
which means that a molecule leaving one face of
the simulation box re-enters at the opposite face.
MEMPHYS 9

10. Polymer Micelle Self-assembly
PEO-PEE diblocks in water:
600 PEO30PEE40 polymers
68 PEO30PEE08 polymers
(water invisible)
Box = 35 x 35 x 105 nm3
Time = 8 µsec
Simulation took 66 cpu-days
Self-assembly is a generic property of amphiphiles: different types of aggregate
are formed depending on: molecular size, ratio of philic to phobic segments, etc.

11. Nanoparticle Self-assembly
216 discoidal nanoparticles (blue)
in a Topo /water mixture (7 mM)
4764 Trioctylphosphine (Topo,
red/orange) molecules (157 mM))
(Water invisible)
Box = (36 nm)3
Simulation took 7 cpu-days
Nanoparticle surface is functionalised to bind to Topo headgroup; tails are
hydrophobic

12. Vesicles
Problem of scale:
Vesicle area ~ D2
Vesicle volume ~ D3
D = vesicle diameter ~50-500 nm
T = membrane thickness ~ 5 nm
For realistic vesicle/cell sizes, we
need D/T ~ 10-2000. This requires
~800,000 beads for 50 nm vesicle
simulation (D/T = 10).
A 10 µm cell simulation needs
> 1,000,000,000 beads.
Current limit is ~ 3,000,000. 9000 lipids in whole membrane; 546 in patch
Identical molecular architecture, but different lipid
types repel creating a line tension around the patch
MEMPHYS 12

13. DPD “State of the Art”
Applications
Polymeric fluids on ~50 nm length scale / microseconds
Vesicle fusion ~ 100 nm / microseconds
Nanoparticle-membrane interactions: tens of nanoparticles
and 50 nm membrane patches
Requirements
½ kB per bead of RAM required
1010 bead-steps per cpu-day
System size limit is ~3 million particles on single processor:
Single fusion event requires ~ 1 cpu-week
MEMPHYS 13

14. Future Requirements
Applications
Rational design of drug delivery vehicles
Toxicity testing of < 1 µm particles for diagnostics
Cell signalling network: receptors, membrane,
cytoskeleton, proteins
Scales
We need: 1 nm – 10 µm, ns – ms
We need at least 3 billion particles for a (1 µm)3 run
(1 µm)3 for 10 µs requires 274 cpu-years on a single
processor: on 1000 nodes with a factor of 1000 speedup,
this becomes 0.1 cpu-day and will create ~500 GB per run
Hardware/Software
Multi-scale model of a cell signalling
1000 commodity, Intel Woodcrest processors;
network:
fast interconnects; database to hold 100 TB data;
XML-based simulation markup language to tag, archive
R1 Dissipative Particle Dynamics
and re-use simulation results;
R2 Brownian Dynamics
automated model phase space search
R3 Differential equations
MEMPHYS 14

15. DPD algorithm: Basics
Particle based: N particles in a box, specify ri(t) and pi(t), i = 1…N.
Mesoscopic: Each particle represents a small volume of fluid with
mass, position and momentum
Newton’s Laws: Particles interact with surrounding particles;
integrate Newton’s equations of motion
Three types of force exist between all particles:
•Conservative FCij(rij) = aij(1 – |rij|/r0)rij / |rij|
FDij(rij) = – γij(1 – |rij|/r0)2(rij.vij) rij / |rij|2
•Dissipative
•Random FRij(rij) = (1 – |rij|/r0)ζijrij / |rij|
forces are soft, short-ranged (vanish beyond r0), central, pairwise-additive,
and conserve momentum locally.
MEMPHYS 15

16. DPD algorithm: Forces
•Conservative FCij(rij) = aij(1 – rij/r0)rij / rij
•Dissipative FDij(rij) = – γij(1 – rij/r0)2(rij.vij) rij / rij2
•Random FRij(rij) = (1 – rij/r0)ζijrij / rij
Conservative force gives particles an identity, e.g. hydrophobic
Dissipative force destroys relative momentum between
pairs of interacting particles
Random force creates relative momentum between pairs of
interacting particles: <ζij (t)> = 0, < ζij (t1) ζij(t2)> = σij2δ(t1-t2), but
note that ζij (t) = ζji (t).
MEMPHYS 16

17. DPD algorithm: Bonds
DPD Polymers are constructed by tying particles together with
a quadratic potential (Hookean spring): the force law is
F(rii+1) = -k2(| rii+1 | - ri0) rii+1 /| rii+1 |
with i,i+1 representing adjacent particles in polymer. Note that k2,r0
may depend on the particle types.
Hydrocarbon chain stiffness may be included
via a bending potential
i j
V(ijk) = k3(1 - cosφijk)
With ijk representing adjacent triples of beads.
k
Again, k3 may depend on particle types.
MEMPHYS 17

18. Vesicle Fusion in Cells
(Scales et al.
Nature 407:144-
146 (2000)).
Synaptic vesicles are guided to the pre-synaptic membrane by “motors”
moving along filaments; they are then held by SNARE proteins in close
proximity to the target membrane.
SNAREs hold the vesicle
close to the membrane
and promote fusion
(Knecht & Grubmueller,
Biophys. J. 84:1527-
1547(2003)).
MEMPHYS 18

19. Fusion Protocol: Tension
Create bilayer and vesicle
under tension
30 nm
50 nm
position them close
together and let evolve
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20. Fusion Run
Vesicle has 5887 lipids; membrane has 5315 in a box (50 nm)3 for 640 ns
20

21. Fusion Run
Vesicle has 6000 lipids; membrane has 3600 in a box (42 nm)3 for 3.2 µs
Lipid headgroup/tail interactions modified to produce a “cone-like” lipid.
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22. Morphology Diagram
Bilayer and vesicle
lipids: H3(T4)2
Relaxed Nves = 6542
Relaxed Nbil = 8228
43 successful fusion
events
out of 92 attempts
MEMPHYS 22

23. Tense Fusion Summary
Fusion occurs (within 2 microsec) near the stability limits of
the aggregates for this parameter set
Our new parameter set shows that flip-flop of lipids from
vesicle to planar membrane is one of two time-scales: there
are two barriers to fusion:
Transfer of vesicle lipids to planar membrane
Rearrangement of disordered contact zone into single
membrane which subsequently ruptures
Shillcock and Lipowsky, Nature Mat. 4:225 (2005)
Grafmueller, Shillcock and Lipowsky, PRL 98:218101 (2007)
MEMPHYS 23

24. Fusion Proteins in vivo
SNARE proteins present in both membranes pull them together
and drive the formation of the fusion pore
But… what do they actually do? Force, torque, displacement…?
Do they pull the pore open or prevent it closing?
MEMPHYS 24

25. Fusion Proteins in silico
Lipid tail beads are
polymerised into “rigid”
cylinders, of radius r, that
span the membranes in a
circle of radius Rp
An external force, of
magnitude Fext, is applied
to pull the barrels apart
radially
MEMPHYS 25

26. Proteins in Fusion
Transmembrane
proteins can exert
forces on the
bilayer
(McNew et al.,
J. Cell. Biol.
150:105 (2000))
See also Venturoli et al, Biophys. J. 88:1778 (2005)
MEMPHYS 26

27. Protein-Induced Fusion Protocol
Define 6 barrels per membrane: e.g., r = 1.5 a0, Rp = 6 a0
Specify the external force magnitude and direction
Measure the time at which the pore first appears and how
large it grows (Fusion time definition: time from when Fext > 0
to when pore diameter is > a few amphiphile diameters)
Shillcock and Lipowsky, J. Phys. Cond. Mat. 18: S1191 (2006)
MEMPHYS 27

28. Typical Fusion Event
Box = 100 x 100 x 42 nm3 28,000 BLM amphiphiles
3.2 x 106 beads in total 5887 Vesicle amphiphiles
MEMPHYS 28

29. Dependence on Force
1
8000
2
7000
6000 3
4 runs per applied force
Work 5000 4
Duration between 40 ns and 64 ns
Done 4000
Barrels move ~ 8nm
(4 x their diameter) /kT 3000
If force is too small, no pore appears
2000
1000
0
214 171 150
External Force
/pN per barrel
NB. Work done is for all 12 barrels
MEMPHYS 29

30. Nanoparticles and Endocytosis
“Rigid” nanoparticles are
constructed by tying beads
together with Hookean
springs giving a “polymerised”
surface whose stiffness can be
modulated by varying the
spring constant
Patches created by changing
selected bead interactions
Star polymers and
PEG-ylated lipids are normal
DPD molecules
MEMPHYS 30

31. Nanoparticles in Bulk
Proteins are bulky, “rigid”
nanoparticles (NP) with sticky patches.
What happens if we place them
In bulk water?
Here are 18 pentagons (shaped like a
protein produced by Shigella
bacterium), floating in water;
The edge and surfaces of each NP
Are hydrophobic.
MEMPHYS 31

32. Nanoparticles near a Membrane
What happens if the NPs can interact
with a nearby membrane?
Here are 9 Shigella proteins floating in
water near a fluctuating membrane.
The surfaces of each NP are
functionalised to adhere to the lipid
headgroups, and to aggregate with
each other.
First, the NPs adhere and slowly diffuse along the surface, next they
discover that by aligning in a chain, the membrane can maintain its
fluctuations in 1 dimension, and so increase its entropy.
MEMPHYS 32

33. Nanoparticle Budding
How can material pass
through a membrane without
rupturing it?
Some viruses enter a cell by a
fusion process that involves them
being enveloped in membrane from
the target cell.
Q What shape of nanoparticle allows it
to be enveloped most readily?
Here, two rigid nanoparticles are placed near a membrane containing
two patches to which the NPs are attracted. The patch lipids are
slightly repelled from the surrounding membrane lipids, and the NPs
adhere to the patches. The combination of adhesion energy and line
tension around the patches drives the budding process.
MEMPHYS 33

34. Endocytosis
How do we construct a coated nanoparticle (NP) in a simulation?
(Initial state assembly)
NP approaches membrane and cross-links receptors (active binding)
Receptors undergo conformational change (modify interactions)
NP is internalised in a vesicle (curvature-induction, budding off)
NP-vesicle modifies signalling response (???)
Experimental questions to answer
What selects the NP size and shape that has greatest effect on receptor
internalisation? (range is 2 – 100 nm in Jiang et al.)
How does the NP surface density of ligands influence receptor response?
What influence does the inplane diffusion of receptors have?
Nanoparticle-mediated cellular response is size-dependent
Jiang et al, Nature Nanotechnology 3:145 (2008)

35. Proteins/nanoparticle
GNP Size / nm
Proteins per Nanoparticle
Surface protein density / nm-2

36. Polymer-coated nanoparticle
Encode self-assembly in polymer’s interactions:
H1-[ B B B S6 B B ]-T1
109 comb polymers; hydrophobic
backbone and hydrophilic sidechains
Spherical nanoparticle with
hydrophobic surface
Apply forces to arrange the
polymers so that they coat the NP
MEMPHYS 36

37. Coated Nanoparticles
We want to make Quantum Dots that consist of a rigid core that is coated by
layers of functional polymers: but how do we wrap the core with the polymers?
5 nm diameter core 5 nm diameter core
25 coat molecules 64 coat molecules
coat = Comb polymer -(B B B (S) B B )8 -
By applying succesive coats we can build up a structured QD
MEMPHYS 37

38. Nanoparticle Bulk Diffusion
4 polymerised (solid) spheres with
100% hydrophilic surface
Box = (25 x 25 x 12.5 nm)3
0.02M HT6 surfactant
Spheres diffuse in solvent, as surfactants micellize
MEMPHYS 38

39. Quantifying Diffusion of Spheres in Bulk Solvent
Mean square displacements (MSD) for 4 spheres (R/a0 = 2) in a (32 a0)3 box:
averaging over several trajectories gives more accurate results.
MEMPHYS 39

40. Stokes’ Law
R = 2 data from
4 spheres in a 323 box
(1 trajectory / 5 cpu-days)
R = 4 data from
1 sphere in a 483 box
(1 trajectory / 17.5 cpu-days)
Fitting R = 4 data from 200-500,000 Fitting the R = 2 data from 200–500,000
and fixing the slope to zero yields: and fixing the slope to zero yields:
Intcpt. = 0.0005 +/- 4.10-6 Intcpt. = 0.0011 +/- 2.10-6
We get D = constant / Radius
MEMPHYS 40

41. Work in progress
• Construct a 2 – 100 nm polymer-coated nanoparticle as QD mimic;
several layers of coat required – polymer architecture, surface
coverage and QD shape are control parameters
• Construct a model plasma membrane with diffusing receptors that
oligomerize; QDs that can bind to the membrane and occlude
receptors; measure signalling pathway
• Parallel code to allow 50 nm particles and (500 nm)2 membrane
containing receptors, signalling apparatus, …
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42. Conclusions
“the limits of your language are the limits of your world”
Wittgenstein
Computer simulations provide a language for describing dynamical
complex systems with (almost) unlimited control
DPD captures processes cheaply (calibration of parameters is time-
consuming); experimentally invisible data are accessible on 100
nm/10 µs time-scales: parallel code can reach 1 µm and milliseconds.
We can observe molecular rearrangements during cellular processes,
e.g., fusion, endocytsosis,…; we can test hypotheses about
interactions and function; build toy models and compare their
predictions to experimental systems; all more cheaply than in a wet lab.
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