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Computational Cell Modeling Julian C Shillcock MEMPHYS Source:

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1 Computational Cell Modeling Julian C Shillcock MEMPHYS Source:

2 MEMPHYS2 Structure of talk Amphiphiles, Membranes and Self-Assembly Vesicles, Fusion & Nanoparticles Requirements and Challenges Summary What are the organizational and dynamic properties of membranes at a molecular level? How can we simulate nanoparticle motion on cellular length scales?

3 MEMPHYS3 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

4 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 scale* Fusion event (0.32 μ sec. ) 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.

5 MEMPHYS5 DPD algorithm: Basics Particle based:N particles in a box, specify r i (t) and p i (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 F C ij (r ij ) = a ij (1 – |r ij |/r 0 )r ij / |r ij | Dissipative F D ij (r ij ) = –  ij (1 – |r ij |/r 0 ) 2 (r ij.v ij ) r ij / |r ij | 2 Random F R ij (r ij ) = (1 – |r ij |/r 0 ) z ij r ij / |r ij | forces are soft, short-ranged (vanish beyond r 0 ), central, pairwise-additive, and conserve momentum locally.

6 MEMPHYS6 DPD algorithm: Forces Conservative F C ij (r ij ) = a ij (1 – r ij /r 0 )r ij / r ij Dissipative F D ij (r ij ) = –  ij (1 – r ij /r 0 ) 2 (r ij.v ij ) r ij / r ij 2 Random F R ij (r ij ) = (1 – r ij /r 0 )  ij r ij / r ij 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: = 0, =  ij 2 d (t 1 -t 2 ), but note that  ij (t) =  ji (t).

7 MEMPHYS7 DPD algorithm: Bonds DPD Polymers are constructed by tying particles together with a quadratic potential (Hookean spring): the force law is F(r ii+1 ) = -k 2 (| r ii+1 | - r i0 ) r ii+1 /| r ii+1 | with i,i+1 representing adjacent particles in polymer. Note that k 2,r 0 may depend on the particle types. Hydrocarbon chain stiffness may be included via a bending potential V(ijk) = k 3 (1 - cos  ijk ) With ijk representing adjacent triples of beads. Again, k 3 may depend on particle types. i j k

8 MEMPHYS8 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

9 Wormlike Micelle Self-assembly Two lipid types in water: 379 H 2 T 5 (long tail) 379 H 2 T 2 (short tail) (water invisible) Box = 30 x 30 x 30 nm 3 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.

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

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

12 MEMPHYS12 Vesicles Problem of scale: Vesicle area ~ D 2 Vesicle volume ~ D 3 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 patch Identical molecular architecture, but different lipid types repel creating a line tension around the patch

13 MEMPHYS13 Typical Fusion Event 28,000 BLM amphiphiles 5887 Vesicle amphiphiles Box = 100 x 100 x 42 nm 3 3.2 x 10 6 beads in total

14 MEMPHYS14 Nanoparticle Self-Assembly 64 NPs (~ 4nm) with 2 hydrophobic patches in (invisible) solvent

15 MEMPHYS15 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.

16 MEMPHYS16 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 then rolling around until they enter by endocytosis. B) Enzymes can bind to a surface and 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.

17 MEMPHYS17 Adsorption Kinetics Plot of the Z coordinate of the particles’ CM versus time. From this data, we can extract the Fraction of time a particle is bound, And use this to calibrate the interaction parameters. Two particles adhere almost completely, and two bind/unbind several times

18 MEMPHYS18 Filament-Coated Membrane

19 MEMPHYS19 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 10 10 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

20 MEMPHYS20 Future Requirements Applications Rational design of drug delivery vehicles Toxicity testing of < 1  m particles for diagnostics Cell 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/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 Multi-scale model of a computational cell: R1 Dissipative Particle Dynamics R2 Brownian Dynamics R3 Differential equations

21 MEMPHYS21 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  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 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; 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

22 22 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

23 MEMPHYS23 Nanotubes Tubes (~ 30 x 6 nm and 30 x 3.5 nm) in (invisible) solvent

24 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 are hydrophobic (more movies at

25 MEMPHYS25 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.

26 MEMPHYS26 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.

27 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-dependent Jiang et al, Nature Nanotechnology 3:145 (2008)

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

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