Computational Cell Modeling

Julian C Shillcock MEMPHYSSource: chemistrypictures.org

MEMPHYS 2

Structure of talk

• Amphiphiles, Membranes and Self-Assembly• Vesicles, Fusion & Nanoparticles• Requirements and Challenges• Summary

What are the organizational and dynamic properties of membranesat a molecular level?

How can we simulate nanoparticle motion on cellular length scales?

MEMPHYS 3

Evolution of 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 of interest, observe self-organization and emergent phenomena; we need software engineering tools to do this

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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 scale*

Fusion event (0.32 μsec. ) with DPD ~200 cpu-hours

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

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

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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) = – ij(1 – |rij|/r0)2(rij.vij) rij / |rij|2

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

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

MEMPHYS 6

DPD algorithm: Forces•Conservative FC

ij(rij) = aij(1 – rij/r0)rij / rij

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

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•Random FRij(rij) = (1 – rij/r0)ijrij / 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: <ij (t)> = 0, < ij (t1) ij(t2)> = ij

2(t1-t2), but note that ij (t) = ji (t).

MEMPHYS 7

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,r0 may depend on the particle types.

Hydrocarbon chain stiffness may be included via a bending potential

V(ijk) = k3(1 - cosijk)

With ijk representing adjacent triples of beads.

Again, k3 may depend on particle types.

i j

k

MEMPHYS 8

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

Wormlike Micelle Self-assembly

Two lipid types in water:

379 H2T5 (long tail)379 H2T2 (short tail)

(water invisible)

Box = 30 x 30 x 30 nm3

Simulation took 7 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.

Polymer Micelle Self-assembly

A-B diblock copolymers in (invisible) solvent + dioxane (X, blue) at decreasing concentration: X condenses the B (red) block.

Polymer Micelle Self-assembly

A-B-C block copolymers in solvent + dioxane (X) at (fixed) high concentration: increasing block lengths (MW).

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VesiclesProblem 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 m cell simulation needs > 1,000,000,000 beads.

Current limit is ~ 3,000,000. 9000 lipids in whole membrane; 546 in patchIdentical molecular architecture, but different lipid types repel creating a line tension around the patch

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Typical Fusion Event

28,000 BLM amphiphiles5887 Vesicle amphiphiles

Box = 100 x 100 x 42 nm3

3.2 x 106 beads in total

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Nanoparticle Self-Assembly

64 NPs (~ 4nm) with 2 hydrophobic patches in (invisible) solvent

MEMPHYS 15

Nanoparticle Budding

How 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 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.

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Nanoparticle Adsorption

Can we measure adsorption of nanoparticles to a rigid surface quantitatively?

A) Some viruses enter a cell by first “adsorbing” to its surface and thenrolling around until they enter byendocytosis.

B) Enzymes can bind to a surfaceand act on it

Here, four rigid nanoparticles are placed near a rigid surface containing two hydrophobic stripes to which the ends of the NPs are attracted. Note the cyan NP at back right that (slowly) flips from stripe to stripe. Surfactants also adhere to the stripes by their tails.

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Adsorption Kinetics

Plot of the Z coordinate ofthe particles’ CM versus time.

From this data, we can extract theFraction of time a particle is bound, And use this to calibrate theinteraction parameters.

Two particles adhere almost completely, and two

bind/unbind several times

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Filament-Coated Membrane

MEMPHYS 19

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

Requirements* ½ 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* 2 GHz Xeon with 2 GB of RAM

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

Scales We need: 1 nm – 10 m, ns – ms We need at least 3 billion particles for a (1 m)3 run(1 m)3 for 10 s 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 computational cell:

R1 Dissipative Particle Dynamics R2 Brownian DynamicsR3 Differential equations

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ChallengesNanoparticle ConstructionNeed to construct coated NPs of various sizes: 10-30 nm, at a specified concentration in a fluid environment of given viscosity; vesicles up to 100 nm diameter

Diffusion We need (0.5 m)3 for ~1 ms to measure diffusion coefficients of NPs and granules (100 nm):Need to be able to predict effects of size/shape/surface coating, concentration,…

Model-Based DiagnosisRelative measurements: Use traces from healthy and diseased beta cells, construct a table of diffusion coefficents for NPs of known sizes;

Absolute measurements: Construct a model cell with spheres, filaments, organelles with the size distribution and concentration specified and measure diffusion of NPs of known sizes; polymer-coated NPs; NPs with specific binding to certain inclusions

A predictive computational cell needs to automate the assembly of structures

from nm to microns as we cannot do it by hand

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Summary“the limits of your language are the limits of your world”

Wittgenstein

DPD captures dynamic processes cheaply (calibration of parameters is

time-consuming); parallel code can reach 1/2 m and millisec

Fluid environment includes HD interactions, spatial organization,

crowding, thermal fluctuations, surfaces, filaments, binding

We can predict NP diffusion as function of size/shape/coating,

and measure NP/membrane adhesion and translocation

Reproducing the internal dynamic conditions of a cell is hard; relative

measurements of NP diffusion in exptal conditions is possible

MEMPHYS 23

Nanotubes

Tubes (~ 30 x 6 nm and 30 x 3.5 nm) in (invisible) solvent

Nanoparticle/Surfactant 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 (more movies at www.complexfluidsimulations.com)

MEMPHYS 25

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 26

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.

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)

Need a parallel code to reach length and time scales of interest

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?

Endocytosis: work in progress

Nanoparticle-mediated cellular response is size-dependentJiang et al, Nature Nanotechnology 3:145 (2008)

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