Transcript Document

Computational Cell Modeling
Julian C Shillcock MEMPHYS
Source: chemistrypictures.org
Structure of talk
What are the organizational and dynamic properties of membranes
at a molecular level?
How can we simulate nanoparticle motion on cellular length scales?
• Amphiphiles, Membranes and Self-Assembly
• Vesicles, Fusion & Nanoparticles
• Requirements and Challenges
• Summary
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Evolution of 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; 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 timescale (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: 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|
•Dissipative
FDij(rij) = – gij(1 – |rij|/r0)2(rij.vij) rij / |rij|2
•Random
FRij(rij) = (1 – |rij|/r0)zijrij / |rij|
forces are soft, short-ranged (vanish beyond r0), central, pairwise-additive,
and conserve momentum locally.
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DPD algorithm: Forces
•Conservative FCij(rij) = aij(1 – rij/r0)rij / rij
•Dissipative FDij(rij) = – gij(1 – rij/r0)2(rij.vij) rij / rij2
•Random
FRij(rij) = (1 – rij/r0)zijrij / 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: <zij (t)> = 0, < zij (t1) zij(t2)> = sij2d(t1-t2), but
note that zij (t) = zji (t).
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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
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V(ijk) = k3(1 - cosfijk)
With ijk representing adjacent triples of beads.
Again, k3 may depend on particle types.
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k
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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
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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 aggregate
are 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).
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 mm 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
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Typical Fusion Event
Box = 100 x 100 x 42 nm3
3.2 x 106 beads in total
28,000 BLM amphiphiles
5887 Vesicle amphiphiles
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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.
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Filament-Coated Membrane
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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
* 2 GHz Xeon with 2 GB of RAM
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Future Requirements
Applications
Rational design of drug delivery vehicles
Toxicity testing of < 1 mm particles for diagnostics
Cell signalling network: receptors, membrane,
cytoskeleton, proteins
Scales
We need: 1 nm – 10 mm, ns – ms
We need at least 3 billion particles for a (1 mm)3 run
(1 mm)3 for 10 ms 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
1000 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
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Multi-scale model of a computational
cell:
R1 Dissipative Particle Dynamics
R2 Brownian Dynamics
R3 Differential equations
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Challenges
Nanoparticle Construction
Need 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 mm)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 Diagnosis
Relative 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; polymercoated 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 mm 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
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