Towards a Computational Cell

Julian C Shillcock MEMPHYSSource: chemistrypictures.org

MEMPHYS 2

Structure of talkWhat are the organizational and dynamic properties of membranesat 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 3

Evolution of (Bio-) simulationsPast

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)

PresentResponse – equilibrium properties & perturbations

FutureControl – we want to interact with a system as it evolves, keep only molecular details necessary to create structure on the scales ofinterest, observe self-organization and emergent phenomena

(The Middle Way Laughlin et al., PNAS 97:32-37, 2000)

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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-hoursFusion event using all-atom MD ~500 cpu-years

MEMPHYS 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 amembrane thickness of ~ 4 nmbut form vesicles/cells with diameters from 50 nm to 10 µm; lipids diffuse in ~100 ns but cells divide in ~minutes. Source: chemistrypictures.org

Plasma membrane and transport vesicles are composed of hundreds of types of lipid and protein molecules

MEMPHYS 6

LipidsLipid 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 7

AmphiphilesHow 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

Cylinder Cone InvertedCone

Source: chemistrypictures.org

MEMPHYS 8

Headgroup Size

0.6 0.7 0.8 0.9

-4

-3

-2

-1

0

1

2

3

4

H2C6H3C6

HC6

Amphiphilearchitecture

Apr/Nr02

σr 0

2 /kBT

Amphiphile architecture modulates planar bilayer response to tension

MEMPHYS 9

Bilayer Self-assembly in Water324 lipid molecules in (invisible) water

Hydrophilic headgroup

Simulation NotesWater is present in all movies, but invisibleto 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.

Hydrocarbon tails

Polymer Micelle Self-assembly

PEO-PEE diblocks in water:600 PEO30PEE40 polymers68 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 aggregateare formed depending on: molecular size, ratio of philic to phobic segments, etc.

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 arehydrophobic

MEMPHYS 12

VesiclesProblem of scale:

Vesicle area ~ D2

Vesicle volume ~ D3

D = vesicle diameter ~50-500 nmT = 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 13

DPD “State of the Art”

ApplicationsPolymeric fluids on ~50 nm length scale / microsecondsVesicle fusion ~ 100 nm / microsecondsNanoparticle-membrane interactions: tens of nanoparticlesand 50 nm membrane patches

Requirements½ kB per bead of RAM required1010 bead-steps per cpu-day

System size limit is ~3 million particles on single processor:

Single fusion event requires ~ 1 cpu-week

MEMPHYS 14

Future RequirementsApplicationsRational design of drug delivery vehiclesToxicity testing of < 1 µm particles for diagnosticsCell signalling network: receptors, membrane, cytoskeleton, proteins

ScalesWe 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/Software1000 commodity, Intel Woodcrest processors; fast interconnects; database to hold 100 TB data; XML-based simulation markup language to tag, archive and re-use simulation results; automated model phase space search

Multi-scale model of a cell signalling network:

R1 Dissipative Particle Dynamics R2 Brownian DynamicsR3 Differential equations

MEMPHYS 15

DPD algorithm: BasicsParticle 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|

•Dissipative FDij(rij) = – γij(1 – |rij|/r0)2(rij.vij) rij / |rij|2

•Random FRij(rij) = (1 – |rij|/r0)ζijrij / |rij|

forces are soft, short-ranged (vanish beyond r0), central, pairwise-additive, and conserve momentum locally.

MEMPHYS 16

DPD algorithm: Forces•Conservative FC

ij(rij) = aij(1 – rij/r0)rij / rij

•Dissipative FDij(rij) = – γij(1 – rij/r0)2(rij.vij) rij / rij

2

•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)> = σij

2δ(t1-t2), but note that ζij (t) = ζji (t).

MEMPHYS 17

DPD algorithm: BondsDPD 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,r0may depend on the particle types.

Hydrocarbon chain stiffness may be included via a bending potential

V(ijk) = k3(1 - cosφijk)

With ijk representing adjacent triples of beads.

Again, k3 may depend on particle types.

i j

k

MEMPHYS 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)).

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Fusion Protocol: Tension

Create bilayer and vesicle under tension

30 nm

50 nm

position them close together and let evolve

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Fusion Run

Vesicle has 5887 lipids; membrane has 5315 in a box (50 nm)3 for 640 ns

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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.

MEMPHYS 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 23

Tense Fusion SummaryFusion 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 membraneRearrangement 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 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 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 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 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 28

Typical Fusion Event

Box = 100 x 100 x 42 nm3

3.2 x 106 beads in total28,000 BLM amphiphiles5887 Vesicle amphiphiles

MEMPHYS 29

Dependence on Force

4 runs per applied forceDuration between 40 ns and 64 nsBarrels move ~ 8nm

(4 x their diameter)If force is too small, no pore appears

010002000300040005000600070008000

Work Done /kT

214 171 150External Force /pN per barrel

1234

NB. Work done is for all 12 barrels

MEMPHYS 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 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 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 33

Nanoparticle BuddingHow can material pass through a membrane withoutrupturing 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 itto 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 areslightly 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.

EndocytosisHow 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 answerWhat 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-dependentJiang et al, Nature Nanotechnology 3:145 (2008)

Proteins per NanoparticleP

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GNP Size / nm

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/ n

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MEMPHYS 36

Polymer-coated nanoparticleEncode 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 37

Coated NanoparticlesWe 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 core25 coat molecules

5 nm diameter core64 coat molecules

coat = Comb polymer -(B B B (S) B B )8 -

By applying succesive coats we can build up a structured QD

MEMPHYS 38

Nanoparticle Bulk Diffusion4 polymerised (solid) spheres with100% hydrophilic surfaceBox = (25 x 25 x 12.5 nm)3

0.02M HT6 surfactant

Spheres diffuse in solvent, as surfactants micellize

MEMPHYS 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 40

Stokes’ Law

R = 2 data from 4 spheres in a 323 box

(1 trajectory / 5 cpu-days)

R = 4 data from1 sphere in a 483 box

(1 trajectory / 17.5 cpu-days)

Fitting R = 4 data from 200-500,000and fixing the slope to zero yields:

Intcpt. = 0.0005 +/- 4.10-6

Fitting the R = 2 data from 200–500,000 and fixing the slope to zero yields:

Intcpt. = 0.0011 +/- 2.10-6

We get D = constant / Radius

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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 thatoligomerize; 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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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.