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Department of Chemical and Bimolecular Engineering
CHEN90023 Chemical Engineering Research Project
Simulation of the Trans...
ii
Summary
The effect of modifications made to Yan and Yu’s existing dissipative particle dynamics (DPD) model
of an inter...
iii
Contents
Summary.........................................................................................................
4
1.0 Introduction
Due to their complex nature, much of the science surrounding forces acting between nanoparticles is
not...
5
endocytic morphology could be replicated by these modifications as it has been shown that PAMAM
dendrimers and other par...
6
aij denotes the soft-core repulsion parameter between particles i and j and is related to the
compressibility of interac...
7
Bond forces:
Dendrimers and lipids are constructed by tying beads together using Hookean springs with the
potential:
(12...
8
The electric field is solved according to Eq. 16:
(16)
• ε, εr and εo, denote the dielectric permittivity of the medium,...
9
2.2 Dendrimers
Dendrimers are snowflake-like macromolecules (refer to Figure 2). They have defined controllable
structur...
10
2.4 Transmembrane Transport:
The plasma membranes of cells serve to separate the intracellular cytoplasm from the extra...
11
Different parameters affect the ability of nanoparticles to pass through the cellular membranes
including, size, shape ...
12
3.0 Literature Review
In recent years, multiple studies have modelled nanoparticle lipid bilayer interactions using com...
13
The advantages and disadvantages of other coarse-graining techniques have been discussed at
length.33
Groot and Madden ...
14
3.3 Ligand-Receptor Modelling
The use of DPD to model endocytosis is still in its infancy. Until recently, the simplici...
15
4.0 Materials and General Methodology
The DPD simulation was constructed using interacting coarse-grained molecules lar...
16
A single G5 PAMAM dendrimer was subsequently introduced into the simulation box above the
bilayer. For illustrative pur...
17
5.0 Results and Discussion
5.1 Self-Assembly of the Modified Lipid Bilayer
In an attempt to ensure self-assembly and an...
18
Figure 14: Bilayer bead density by displacement from bilayer beads’ average z-coordinate (refer to Appendix 2).
As illu...
19
5.2 Comparison with Yan and Yu’s Results
To further test the stability of the modified DPD model, attempts were made to...
20
Table 2: Historic soft-core repulsion parameters used in the combined simulation.1, 2
Past simulation of the G5 PAMAM d...
21
5.3 Simulation of the System with Dendrimer Heads modelled as Ligands
Despite the non-steady-state morphological discre...
22
Further data is required if the observed trends of increasing radii of gyration and asphericity with the
decrease in th...
23
Figure 22: Penetration data bounded by linear regression confidence bands.
In all simulations the dendrimers were able ...
24
compounded by these unstable regions crossing the boundaries of the simulation environment and
transporting to the othe...
25
Figure 25: Radii of gyration of simulated dendrimer-bilayer systems after approximately 245 timesteps.
As previously di...
26
As discussed, the dendrimer that did not bind to the bilayer was axially stretched by the counterions.
For αRL=0 most o...
27
A phase table describing the morphologies for different soft-core repulsion parameters was
generated:
Table 3: Phase ta...
28
6.0 Further Discussion
6.1 Known Shortcomings of the Model
Given the simplicity of the simulated membrane a true repres...
29
Alternatively, the N-varied DPD method maintains a constant membrane tension by adding and
removing water and lipids fr...
30
6.2 Relationship of CME to Simulations
Using this receptor model, encapsulation of the dendrimer was able to be simulat...
31
into the DPD model have been made or discussed (e.g. isolated dendrimer size, lipid diffusion, hole
generation, hole si...
32
7.0 Conclusions
Incorporation of a receptor-simulating lipid into the bilayer and subsequent tuning of soft-core
repuls...
33
8.0 Suggestions for Further Work
Future experiments require access to appropriate computing resources similar to the cl...
34
If CME is not observed following the simple revisions this model could still be extended to simulate a
larger interacti...
35
9.0 References
1. Yan, L.-T. and X. Yu, Enhanced Permeability of Charged Dendrimers across Tense Lipid Bilayer
Membrane...
36
21. McNerny, D.Q., P.R. Leroueil, and J.R. Baker, Understanding Specific and Nonspecific
Toxicities: A Requirement for ...
37
43. Venturoli, M., et al., Mesoscopic Models of Biological Membranes. Physics Reports, 2006.
437(1–2): p. 1-54.
44. Ler...
38
10.0 Appendices
Appendix 1: Simulation Soft-core Repulsion Parameters
Table 4: Historic soft-core repulsion parameters....
39
Table 6: Soft-core repulsion parameters for interacting bilayer-dendrimer system. Starred parameters were introduced
no...
40
Table 8: Revised soft-core repulsion parameters for interacting bilayer and dendrimer.
Lipid
Head
(+1)
Lipid
Head
(-1)
...
41
Appendix 2: MATLAB Code for Bilayer Bead Density Profile
% Bilayer Bead Density by Displacement from Average z-coordina...
42
y=y';
%histogram smoothing
smooth = sgolayfilt(y,3,29);
subplot(2,1,1)
plot(x, smooth,'linewidth', 2)
axis([-5 5 0 3.5]...
43
Appendix 3: Asphericity
Asphericity of the dendrimer can be calculated from the following equations:
(20)
Where I1 and ...
44
ytot = ytot+y;
end
yavg= ytot/(numel(b)/4);
for i=1:numel(b)/4;
z = b(3,i);
ztot = ztot+z;
end
zavg= ztot/(numel(b)/4);...
45
Appendix 4: Radius of Gyration
The radius of gyration is determined by the following equation:
(25)
Where the summation...
46
%squared displacement from centre of gravity squared
dispsqutot=[0;0;0];
disp = [0;0;0];
for i=1:numel(b)/4;
for j=1:3
...
47
Appendix 5: Penetrability
Penetrability was defined as the fraction of dendrimers that were below the average z-coordin...
48
z = (3,i);
ztot = ztot+z;
end
zavg= ztot/(numel(d)/4)
test=ones(1,numel(b)/4);
%test if dendrimer below middle of bilay...
49
Appendix 6: Confidence Bands
Data file:
aRL fp
15 0.1109
15 0.1327
15 0.1426
15 0.1485
15 0.1089
0 0.1386
0 0.1762
0 0....
50
Appendix 7: Intermediate Data
Table 9: Rg data for αTU=28
αRLTimestep 98000 99000 100000 101000 102000 μ σ
15 3.8E+00 3...
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CHEN90023 Lachlan Russell 389374

  1. 1. Department of Chemical and Bimolecular Engineering CHEN90023 Chemical Engineering Research Project Simulation of the Transmembrane Transport of Poly(amidoamine) Lachlan Russell 389374 Supervisors: Prof. Scales, Assoc. Prof. Li-Tang Yan
  2. 2. ii Summary The effect of modifications made to Yan and Yu’s existing dissipative particle dynamics (DPD) model of an interacting generation five (G5) Poly(amidoamine) (PAMAM) dendrimer and a lipid bilayer were examined.1, 2 The primitive bilayer was modified to include another double-chain lipid to simulate the effect of ligand receptors on the transmembrane transport of dendrimer structures. The structures formed by the interacting lipid bilayer and G5 dendrimer were analysed to see if they resembled morphologies likely to be involved in clathrin-mediated endocytosis (CME). The effect of changes to DPD soft-core repulsion forces acting between the dendrimer surface beads and the receptor beads, as well as the internal dendrimer structure and the hydrophobic lipid tails, were analysed. It was found that self-assembly of the bilayer structure was not compromised when converting a quarter of the simulated lipid molecules into receptor-simulating lipids. Binding and penetration were found to be largely affected by the inclusion of an attractive interaction between receptors and ligands coupled with changes to the magnitude of the repulsion between the bilayer lipid tails and the internal structure of the dendrimer. Encapsulation of the dendrimer by the bilayer was observed for a particular set of soft-core repulsion parameters. The morphology closely matched that of a phagocytic mechanism. On the simulated length scale such a result was unexpected. Further refinement of the model is required if CME, as it is currently understood, is to be simulated and the unobservable dynamics conjectured. Due to computational constraints, many of the results presented are preliminary in nature.
  3. 3. iii Contents Summary....................................................................................................................................................................................ii 1.0 Introduction.........................................................................................................................................................................4 2.0 Relevant Theory...................................................................................................................................................................5 2.1 Dissipative Particle Dynamics (DPD) ...............................................................................................................................5 2.2 Dendrimers .....................................................................................................................................................................9 2.3 Lipid Bilayers ...................................................................................................................................................................9 2.4 Transmembrane Transport: ..........................................................................................................................................10 3.0 Literature Review...............................................................................................................................................................12 3.1 A Case for Dissipative Particle Dynamics (DPD) ............................................................................................................12 3.2 Using Dissipative Particle Dynamics to Model Transmembrane Transport ..................................................................13 3.3 Ligand-Receptor Modelling...........................................................................................................................................14 3.4 Aim of Research ............................................................................................................................................................14 4.0 Materials and General Methodology.................................................................................................................................15 5.0 Results and Discussion.......................................................................................................................................................17 5.1 Self-Assembly of the Modified Lipid Bilayer..................................................................................................................17 5.2 Comparison with Yan and Yu’s Results .........................................................................................................................19 5.3 Simulation of the System with Dendrimer Heads modelled as Ligands........................................................................21 5.3.1 Initial Attempts to Generate Clathrin-mediated Endocytic Morphology ..............................................................21 5.3.2 Further Attempts to Generate Clathrin-mediated Endocytic Morphology ...........................................................24 6.0 Further Discussion .............................................................................................................................................................28 6.1 Known Shortcomings of the Model...............................................................................................................................28 6.2 Relationship of CME to Simulations..............................................................................................................................30 6.3 The Notable Shortage of Empirical Data.......................................................................................................................30 7.0 Conclusions........................................................................................................................................................................32 8.0 Suggestions for Further Work............................................................................................................................................33 8.1 Simple Immediate Revisions to the Current DPD Model ..............................................................................................33 8.2 Extension of Results......................................................................................................................................................33 9.0 References .........................................................................................................................................................................35 10.0 Appendices ......................................................................................................................................................................38 Appendix 1: Simulation Soft-core Repulsion Parameters ...................................................................................................38 Appendix 2: MATLAB Code for Bilayer Bead Density Profile...............................................................................................41 Appendix 3: Asphericity ......................................................................................................................................................43 Appendix 4: Radius of Gyration ..........................................................................................................................................45 Appendix 5: Penetrability....................................................................................................................................................47 Appendix 6: Confidence Bands ...........................................................................................................................................49 Appendix 7: Intermediate Data...........................................................................................................................................50
  4. 4. 4 1.0 Introduction Due to their complex nature, much of the science surrounding forces acting between nanoparticles is not well understood.3 Dissipative particle dynamics (DPD) can be used to effectively model the mesoscopic (10 nm - 100 mm) dynamics of interacting particles (refer to Figure 1).4, 5 DPD attempts to incorporate the essence of many forces believed to be significant on this scale. The approximations utilised by DPD methods allow for mesoscale simulation of many physical interactions that otherwise could not be run at acceptable speeds. Interactions between synthetic dendrimers and cellular membranes are significant due to the growing use of dendrimers in medicine.6 These mesoscale interactions need to be studied in order to synthesise drugs which allow for specific transmembrane transport whilst moderating cytotoxicity.7 DPD has been used to simulate dendritic particles interacting with lipid bilayers.1, 2, 8 Such simulations can model the behaviour of different types of dendrimers reducing the need for costly drug testing by synthesis and administration. Figure 1: Schematic showing coarse-graining used to model the mesoscale. 9 Interactions between lipid bilayers and dendrimers have been studied extensively. However, modern experimental techniques are currently unable to answer many fast-acting molecular-level mechanistic questions.10 It is hoped, given DPD’s success with modelling other mesoscopic phenomena, that simulations can provide additional insight until such a time when observational techniques are sufficiently advanced to provide a more detailed description of interaction mechanisms and morphology. This project aims to extend previous DPD models of interacting lipid bilayers and dendrimers developed by Yan and Yu.1, 2 Whilst amphiphilic lipid molecules comprise the bulk of cell membranes, transport mechanisms often involve receptor proteins which are also present in the structure of the membrane. The simulations performed incorporate primitive receptor-type molecules into Yan and Yu’s simple bilayer. Such modifications have been utilised in other simulations however the effect of such changes on the transmembrane transport of dendrimers such as poly(amidoamine) (PAMAM) have yet to be rigorously studied.11-13 The effects of this change, along with the variation of DPD soft- core repulsion parameters, was compared with Yan and Yu’s experiments, other simulation studies and mechanisms proposed by biologists. Specifically, it was assessed whether clathrin-mediated
  5. 5. 5 endocytic morphology could be replicated by these modifications as it has been shown that PAMAM dendrimers and other particles of similar length scales are internalised via this mechanism.14, 15 The theory behind dissipative particle dynamics will first be introduced followed by a brief description of dendrimers, cell membranes and transmembrane transport. Subsequently, a review of literature concerning the use of computer simulation to model these transport processes will be presented. The details of the modified DPD model will be covered and the results of simulations examined. Lastly, conclusions and suggestions for further research will be presented. 2.0 Relevant Theory 2.1 Dissipative Particle Dynamics (DPD) Hoogerbrugge and Koelman developed a coarse-grained/beaded simulation method combining features of molecular dynamics (MD) and lattice-gas automata. Dissipative particle dynamics simulates more quickly than MD and is more flexible than lattice-gas automata schemes.16 A reformulation of DPD was developed and applied to the modelling of bilayers by Groot and his colleagues.17, 18, 19 The simulation equations adopted by Yan and Yu are largely consistent with the aforementioned theory developed by Groot; these equations were utilised for the sake of consistent simulation.1 Newton’s equations of motion are solved for interacting particles: (1) For simplicity, the mass of particles are set to equal 1 and hence force equals acceleration. Forces act within a dimensionless cut-off radius (rc) of 1. The bilayer-dendrimer DPD system is governed by five forces: Conservative, Dissipative, Random, Bond and Electrostatic. The forces acting on particle i are summed at each timestep: (2) Conservative force: (3) (4), (5), (6)
  6. 6. 6 aij denotes the soft-core repulsion parameter between particles i and j and is related to the compressibility of interacting particles.9 The repulsion parameter can often be related to Flory-Huggins χ parameters in the instance where water is used as the solvent: 𝝌𝒊𝒋 ≈ 𝜶 𝒊𝒋−𝜶 𝒊𝒊 𝟑.𝟐𝟕 (7) • αii =25 is used in general DPD systems for all particle types. • αij will be larger than 25 for strong bead-bead repulsion interactions; it will be smaller than 25 where there exists reasons for attraction between two types of beads (e.g. due to electrostatic or entropic effects). For many of the cross terms, estimations are tested and refined based on qualitative knowledge and simulation results. A complete set of accurate Flory-Huggins χ-parameters often cannot be generated for complex systems with many types of beads present. Dissipative and Random forces: The dissipative force and the random force act as the heat sink and source, respectively. They are balanced to maintain a Boltzmann distribution and hence create a thermostat. (8) (9) • wD and wR are r-dependent weight functions vanishing for r>rc. For simplicity they are chosen to be: (10) • vij=vi-vj and vi denotes the velocity of bead i. • ξij is a random number with zero mean and unit variance. • The noise amplitude, σ, is fixed at σ =3. • The forces are connected by Eq. (11). (11) • This thermostat conserves linear and angular momentum, resulting in a correct description of hydrodynamics.
  7. 7. 7 Bond forces: Dendrimers and lipids are constructed by tying beads together using Hookean springs with the potential: (12) For the purposes of simulation: • The spring constant is set to ks=64. • The unstretched length is set to l0=0.5rc. • Hence, the average bond length is fixed. A three-body potential acting between adjacent bead triples models hydrocarbon chain stiffness in the lipids and dendrimer: (13) • The angle φ is defined by Eq. (14) where A, B denote the two bonds connecting beads i-1, i, and i+1: 𝐜𝐨𝐬−𝟏 𝑨∙𝑩 ‖𝑨‖‖𝑩‖ = 𝝓 (14) • The bending constant ka=20 is used for both the lipid and dendrimer. • The preferred angle, φ0, is specified as 00 and 1200 for lipids and dendrimer respectively. From these potentials, corresponding forces can be calculated. Electrostatic force:18 A lattice grid is constructed, over which the electrostatic field is spread. There is a trade-off made between correct representation and fast implementation of the field. Each charged bead has charge proportional to: (15) This charge is calculated for every grid node within a radius Re =1.6rc of the bead and normalised so that the sum of all proximal charged nodes is equal to the charge of the bead.
  8. 8. 8 The electric field is solved according to Eq. 16: (16) • ε, εr and εo, denote the dielectric permittivity of the medium, the vacuum and the water respectively. • 𝜌̅ 𝑒,𝑛 is the averaged charge density. • ∇𝜑 𝑛 is the electric gradient at node n. • Γ =13.87 is the coupling constant corresponding to the Bjerrum length of water at 300K. At each timestep the electrostatic force on an ion can finally be determined by Eq. (17): (17) • Where qi is the charge of the ion. Algorithm: The velocity-Verlet algorithm is iterated for each bead at each timestep: 𝜆 = 0.65 has been found to reduce computational error accumulating whilst allowing for a relatively large timestep to be chosen. In the simulations, the radius of interaction, the bead mass, and the temperature are set such that rc=m=kBT=1. A characteristic time scale can thus be defined: (18)
  9. 9. 9 2.2 Dendrimers Dendrimers are snowflake-like macromolecules (refer to Figure 2). They have defined controllable structures of known physical, chemical and biological properties. Their associated monodispersity and reproducibility make them ideal for biological applications as structure-activity relationships can be more accurately studied.20 They are defined by their core structure and, as additional shells (generations) are added, approximately double in size and number of surface functional groups. The chemical backbone and surface terminal groups have a large effect on the biological properties of dendrimers. Figure 2: Poly(amidoamine) (PAMAM): Structure comprises of repetitively branched amide and amine functional groups.20 Dendrimers are increasingly being used for drug delivery as, when paired with pharmaceutically active compounds, their defined properties can be used to improve drug specificity.21 Dendrimers are capable of delivering drugs by many mechanisms. They can function as micelles which encapsulate hydrophobic drugs; alternatively, the terminal functional groups of the dendrimer can covalently attach to pharmaceutical agents.20 Dendrimers used for biomedical purposes usually have specific ligands attached to their often cationic surface which interact with receptors on the surface of cellular membranes. 2.3 Lipid Bilayers Phospholipids are surfactant molecules with a hydrophilic head and two hydrophobic tails which can self-assemble to form bilayers (refer to Figure 3). Complex lipid bilayers encapsulate cells and many important organelles.2 Figure 3: Bilayer sheet.22
  10. 10. 10 2.4 Transmembrane Transport: The plasma membranes of cells serve to separate the intracellular cytoplasm from the extracellular environment by controlling the transport of molecules across the membrane.23 Endocytosis is one of many types of processes by which cells internalise molecules (refer to Figure 4). It involves the deformation of the cell membrane and, subsequently, smaller membrane-bound carriers are generated.24 Figure 4: Various methods of mass transport across cell membranes.25 Clathrin-mediated endocytosis (CME) is the chief receptor-mediated endocytic pathway.25 Hence, it is the most widely studied endocytic mechanism. First observed by using electron microscopy, CME is vital for such functions as nutrient uptake and receptor signalling.26 Mechanistically it involves proteins with receptor sites binding to specific ligands, triggering the formation of ‘coated pits’ which subsequently become fully enveloped by the cell (refer to Figure 5).24 Figure 5: Clathrin-mediated endocytosis requires specific ligand-receptor binding interactions.27
  11. 11. 11 Different parameters affect the ability of nanoparticles to pass through the cellular membranes including, size, shape and surface chemistry.6 Surface chemistry strongly affects the interactions between nanoparticles and cells.24 Nonspecific binding forces caused by nanoparticle characteristics such as cationic charge and roughness can also promote cellular uptake in conjuction with ligand- receptor interactions.27 Consequently, it is believed that multiple transmembrane transport mechanisms can work simultaneously (refer to Figure 6).6 Figure 6: Receptor-mediated uptake with additional nonspecific binding forces.27
  12. 12. 12 3.0 Literature Review In recent years, multiple studies have modelled nanoparticle lipid bilayer interactions using computer simulation.7 This is because, despite extensive research in this area, experimentalists have largely failed to elucidate the kinetic evolutions of these interactions. Dissipative particle dynamics (DPD) and other coarse-graining techniques have been used widely to attempt to model these processes. Many of these studies have concerned themselves with the effects of nanoparticle shape, size and surface chemistry on the internalisation process.7 Few have concerned themselves with seeking to replicate the true complexity of cell membranes instead modelling them simply as lipid bilayer sheets. Lately, attempts have been made to model these membranes in increasingly representative ways. The effect of the inclusion of ligand-receptor type molecules on a simulated bilayer-dendrimer system requires further study in the wake of recent developments.12, 13, 28, 29 3.1 A Case for Dissipative Particle Dynamics (DPD) At the nanoscale, interfacial forces dominate interactions. These include van der Waals, electrostatic, depletion, hydration and hydrophobic forces.24 Compared with atomistic models which seek to incorporate much of the complexity of modern physics, DPD utilises a different force potential between interacting beads. In atomistic models, hard-core potentials such as Lennard–Jones (L–J) potential, which incorporates van der Waals attraction and Pauli exclusion forces are used. In DPD models, a soft-core conservative force is used.9 This soft-core repulsion is important as hard-core modelling causes a ‘caging’ effect, where atoms undergo multiple collisions before any transport occurs.18 Hence, models using soft repulsive spheres allow for the use of larger timesteps and length scales. Figure 7: Schematic illustrating the hard-core nature of Lennard–Jones potential and the soft-core nature of DPD inter- particle potential.9 The DPD model attempts to compensate for missing physics by incorporating a dissipative force to account for viscous effects; the molecular level randomness, which usually manifests as Brownian motion, is modelled by the random force.9 Both DPD and MD methods can produce correct equilibrium distributions of polymer chains. However, MD simulations do not include hydrodynamic interactions. Conversely, DPD is able to correctly simulate compressible Navier-Stokes behaviour.30, 31 This result is important at the mesoscale. It is currently possible to simulate novel scenarios of small dendrimers interacting with simple lipid bilayers using atomistic MD models; however, in order to generate meaningful results at greater length and time scales, coarse-graining is required.7 Coarse-graining enables significant speeding up of simulations of interacting membranes and particles orders of magnitude larger.22, 32
  13. 13. 13 The advantages and disadvantages of other coarse-graining techniques have been discussed at length.33 Groot and Madden argue that what is chiefly required of a mesoscale physics model is that it can reproduce thermodynamic properties on the relevant length scale. Namely, if the solubilities, liquid compressibilities and the shape of interfaces are correct, then the model correctly represents the physical system for the given length scale.4 Well-conditioned DPD simulations have been shown to pass such tests.4 3.2 Using Dissipative Particle Dynamics to Model Transmembrane Transport The fundamentals and technical subtleties of DPD are well established.4, 18, 19, 34 Traditionally DPD has been used to model polymer-colloid systems where robust experimental results abound.4, 30, 34 Shilcock and Lipowsky were able to formulate self-assembling planar bilayers with experimentally consistent density profiles and lateral stress distributions using the DPD model.22 These models have been shown to describe the phase behaviour of phospholipids with accuracy.33, 35 Following the successful development of thermodynamically robust self-assembling bilayers many have attempted to study their interactions with nanoscale particles. The modelling of the nanoparticle-bilayer interactions have typically involved the use of the polyamidoamine (PAMAM), which is the most commonly used dendrimer in biomedical applications, and a simple bilayer comprised entirely of dipalmitoylphosphatidylcholine (DPPC).21 In addition to DPD, a variety of coarse- grained and atomistic simulation models have been used for this purpose.10, 13, 36, 37 Yan and Yu used the bilayer developed by Shilcock and Lipowsky to model interactions with cationic G5 PAMAM dendrimers. The effect on morphology caused by changes to the soft-core repulsion parameters, especially those concerning the outer-dendrimer hydrophilic components and the inner-dendrimer hydrophobic components, was analysed extensively.2 Cationic dendrimers are known to be cytotoxic.37, 38 By introducing controlled surface tension into the bilayer model, Yan and Yu were able to observe holes forming on the bilayer surface on interaction with PAMAM dendrimers consistent with experimentally observed phenomena.1 They were subsequently able to control the soft-core interaction parameters of the lipids used in the bilayer to modify their shape factor ( 𝜈 𝛼𝑙 < 1) to instead form vesicles of various sizes and surface tensions and again interact these with PAMAM dendrimers.8 An attempt to better model the endocytic process was incorporated by Guo, Mao and Yan.39 Canham-Helfrich theory was applied to incorporate contact and membrane bending energies.40 Figure 8: DPD simulated endocytosis with incorporated membrane elasticity forces.39
  14. 14. 14 3.3 Ligand-Receptor Modelling The use of DPD to model endocytosis is still in its infancy. Until recently, the simplicity of the simulated bilayers was such that meaningful comparisons to cell membranes were tenuous. Most transport across cell membranes occurs with the involvement of receptor proteins. Very little work has been attempted to model clathrin-mediated endocytosis using simulations as it can be difficult to incorporate changes to the bilayer that do not compromise its self-assembling properties. Attempts to circumvent this problem have been made by modifying the surface of interacting dendrimers to have a proportion of surface beads exhibit a greater affinity for the bilayer.13 More direct attempts modelled ligand-receptor interactions by embedding symmetrical plug-like receptors into a bilayer and interacting them with spherical molecules with uniformly anchored ligand sites. CME mechanisms resembling those proposed by biologists were generated.28 Another strategy involved the incorporation of another type of lipid into the bilayer structure with a ligand-receptor headgroup. This method preserved self-assembly.11-13 The addition of ligands to the model has been approached in a variety of ways. Yang and Ma model their interacting particle as a single large bead which interacts with ligand receptors according to a modified Lennard–Jones potential in order to incorporate the attractive force expected between ligands and receptors.29 Other studies have all used different models to incorporate ligands into simulations.11, 12, 29, 41 While many of these studies produce interesting findings they all fail to model the interacting particle as anything more than solid polyhedrons or spheres. 3.4 Aim of Research The aim of this study is to introduce a receptor-simulating double-chain lipid into the bilayer and interact this modified bilayer with more complex particles such as PAMAM dendrimers. It has been shown that PAMAM dendrimers are internalised primarily through clathrin-mediated endocytosis.14, 15 It is hypothesised that these structural changes to the bilayer combined with the balancing of soft- core repulsion parameters can induce clathrin-mediated endocytic (CME) morphology. In order to test this hypothesis a self-assembling bilayer must first be able to be generated. The stability of the DPD model will be evaluated on comparing with Yan and Yu’s previous simulations.1, 2 Only then can it be determined if structures resembling CME can be generated with or without the additional refinement of some soft-core repulsion parameters. Any such refinements should be justifiable either with empirical data or, less ideally, using qualitative arguments. Depending on the strength of these justifications the hypothesis may be able to be supported. Consequently, it could be argued that the simulation results might describe currently unobservable dynamics. If so, the model or its derivatives may be able to assist in the future development of new dendrimers for drug delivery.
  15. 15. 15 4.0 Materials and General Methodology The DPD simulation was constructed using interacting coarse-grained molecules largely consistent with previous work by Yan and Yu.1 The lipid bilayer was constructed by creating a coarse-grained representation of dipalmitoylphosphatidylcholine, the phospholipid commonly used to study lipid bilayers. DPPC consists of a phosphate group, a simple organic molecule choline and two palmitic acid chains. Figure 9: Dipalmitoylphosphatidylcholine.17 The simulation model of DPPC is consists of a hydrophilic head with charge +1 (green), a hydrophilic head with charge -1 (purple) a head without charge (blue) and a hydrophobic tail group (cyan). The separation between the beads was set to 0.5rc. Figure 10: Bead model of DPPC.1 Another type of lipid was introduced into the bilayer structure.11, 12 Primitive receptor-simulating double-chain lipids were created using a similar structure consisting of two uncharged hydrophilic heads (orange), another uncharged head (blue) and a hydrophobic tail group (cyan). The separation between the beads was set to 0.5rc. Figure 11: Bead model of primitive receptor-simulating double-chain lipid. For each of the double-chained lipids introduced into the simulation box (approximately 2360 in total) a random number generator gave the lipid a 25% chance of being a receptor-simulating double-chain lipid (refer to Further Discussion). The DPD simulation was allowed to run in an aqueous environment until a steady bilayer structure self-assembled.
  16. 16. 16 A single G5 PAMAM dendrimer was subsequently introduced into the simulation box above the bilayer. For illustrative purposes, the coarse-grained representation of a G1 PAMAM dendrimer is depicted in Figure 12. Figure 12: Bead model of G1 PAMAM dendrimer.1 The yellow beads represent the hydrophobic inner amine and amide functional groups and are uncharged. The red beads represent the hydrophilic surface amine functional groups and carry a charge of +1. The separation between the beads was set to 0.5rc. Consistent with Yan and Yu’s modelling, counterions were also included in the simulation.2 Each simulation was run for more than 105 timesteps of length Δt=0.02τ in order to ensure accurate temperature control. The approximate area per lipid in a tensionless DPPC membrane is 0.64 nm2 . From this, the cut-off radius (rc) can be estimated to be approximately 0.7 nm. The time unit (τ) can be related to physical time by known inplane lipid diffusion coefficients. Experiments have shown this value to be approximately 5 μm2 /s. Relating this to Yan and Yu’s previous work τ≈7.7ns and, hence, 105 time steps equates to approximately 15 μs.2 Similarly, most of the soft-core repulsion parameters used, (refer to Appendix 1: Table 4), were compiled by Yan and Yu from the literature and, in the absence of sources, refined through qualitative arguments and iterative simulation.1 The introduction of the new receptor-simulating lipids required additional interaction parameters to be generated. The effect of varying some of these parameters, both new and old, was studied in order to attempt to build a model that successfully simulated a bilayer-dendrimer system interacting in a way that mimicked proposed clathrin-mediated endocytic mechanisms (refer to Results and Discussion). The Fortran code used to perform these DPD simulations has been made available (refer also to Theory: Dissipative Particle Dynamics (DPD)). Similarly, the MATLAB programs written to perform subsequent analysis of binary output from these programs can be found in the appendices.
  17. 17. 17 5.0 Results and Discussion 5.1 Self-Assembly of the Modified Lipid Bilayer In an attempt to ensure self-assembly and an even distribution of receptors on the surface of the bilayer; parameters for the new receptor beads were set such that their affinity for the other headgroup beads was equivalent to their affinity for one another (α=25). As per other headgroup beads, the receptor beads were designed to be similarly repelled by the hydrophobic lipid chains (α=40).1 So as to achieve the slight protrusion of these receptors above the bilayer (refer to Figures 5, 6) receptor beads were given smaller repulsion parameters than other headgroup beads to govern their interactions with water (α=15) (refer to Appendix 1: Table 5). Consistent with expectations, for these carefully selected soft-core repulsion parameters, the introduction of the new lipid molecule had no effect on the self-assembling property of the bilayer. It was observed that an even surface distribution of this new coarse-grained molecule was achieved (refer to Figure 13). The total lipid density was determined to be 1.47𝑟𝑐 2 =0.72 nm2 approximately 13% greater than Yan and Yu’s literature value of 0.64 nm2 .1 This minor difference was due to a change to the ‘target lipid density’ input variable in the code used to simulate the self-assembly of the bilayer. The simulation box is continuous in the sense that beads cannot leave the box and are simply transported to the other side if they move across a boundary. The targeted density is achieved by the coded insertion or removal of more lipids once the bilayer structure is relatively stable. Simulation of the steady-state self-assembly of a bilayer required significant computer processing. This minor change to the ‘target lipid density’ was performed at the request of a colleague due to limited computing resources in order to test the effect of such a change on the stability of the bilayer structure. This increase in density lead to slightly less exposure of the lipid tails when compared to the surfaces originally constructed by Yan and Yu.1, 2 This deviation was partially offset by the slight protrusion of receptor beads into the water phase. Figure 13: Self-assembling bilayer produced with 1:3 ratio of receptor-simulating double-chain lipids and DPPC lipids.
  18. 18. 18 Figure 14: Bilayer bead density by displacement from bilayer beads’ average z-coordinate (refer to Appendix 2). As illustrated by Figure 14, consistent with the DPD model, total bead density (solvent included) was centered on 3 beads/nm3 .17 The density distribution for the bilayer beads was largely consistent with previous studies with expected deviation arising from the 1:3 ratio of receptor-simulating double- chain lipids and DPPC lipids.1, 22 The paired receptor beads protruded approximately 0.4 nm further into the water than the two topmost combined headbeads on the DPPC lipids (refer to Figures 10,11). Due to computational constraints it was not possible to revise the receptor-water interactions, lipid- receptor split and lipid density to be consistent with literature (refer to Suggestions for Further Work).
  19. 19. 19 5.2 Comparison with Yan and Yu’s Results To further test the stability of the modified DPD model, attempts were made to replicate results from Yan and Yu’s previous studies.1, 2 The affinity of all lipid headgroup beads for the external dendrimer beads and the internal dendrimer beads were set to αRL=15 and αTU=28 respectively, consistent with Yan and Yu’s soft-core interaction parameters (refer to Appendix 1: Table 6). After 105 timesteps (approximately 15 μs), a contrasting structure was generated: Figure 15: Comparison of dendrimer penetration after 15 μs. Note that the counterions are not shown in both figures; however, their presence is incorporated into both models consistently. Due to the presence of the receptor-simulating lipid in the bilayer structure, this simulation is subtly different to Yan and Yu’s previous work. Figure 15 depicts a bilayer structure which is less planar, with lipids capable of migrating onto the surface of the dendrimer; this was unexpected. The reasons for these discrepancies require further investigation. However, it is worth noting that the bending of the bilayer reduced as the simulation progressed and the migration of lipids onto the dendrimer was observed by Yan and Yu in another simulation which used the same soft-repulsion parameters.2 This finding potentially highlights the effect of DPD’s random force on simulations. The simulated dendrimer was also slightly more spherical, with the asphericity calculated to be As=0.033 compared with Yan and Yu’s figure centred on As=0.055 (refer to Appendix 3); the magnitude of this variation was deemed not to be significant given the dynamic nature of the system. Importantly, the structure of the simulated dendrimer was considerably more open in the modified system with the radius of gyration equal to Rg=3.8 nm compared with Rg=2.7 nm (refer to Appendix 4). The magnitude of this deviance was unexpected. The most likely causes of this deviation are unintentional changes from historical soft-core repulsion parameters that govern the simulation environment of the dendrimer equilibrating in water prior to being introduced into the simulation box with the lipid bilayer. Table 1: Soft-core repulsion parameters governing isolated dendrimer-water system. Solvent Bead (0) Surface Dendrimer Bead (+1) Internal Dendrimer Bead (0) Solvent Bead (0) 25 25 60 Surface Dendrimer Bead (+1) 25 25 30 Internal Dendrimer Bead (0) 60 30 25 (Yan & Yu 2009)
  20. 20. 20 Table 2: Historic soft-core repulsion parameters used in the combined simulation.1, 2 Past simulation of the G5 PAMAM dendrimers in water using the parameters defined in Table 2 yielded compact spherical structures with diameters consistent with empirical measurements.2, 42 Figure 16: Equilibrium configuration of a G5 PAMAM dendrimer in water 2. The effect of the soft-core repulsion parameters changes, especially the reduction of the hydrophobicity of internal dendrimer beads (from 80 to 60), caused the dendrimer to equilibrate to a size approximately 40% larger than desired. Figure 17: Dendrimer introduced into the combined simulations The simulation codes which modelled the dendrimer isolated from the bilayer were not made available and the artifacts were only detected after returning to Australia and hence could not be rectified. In the combined simulation environment the parameters were restored to their original values; however, the dendrimer did not have sufficient time to structurally equilibrate before interacting with the bilayer in its more open form (refer to Suggestions for Further Work). Minor structural variation was likely further compounded by other sources. As previously mentioned, due to the inclusion of random forces in the DPD model an exact morphological match was not expected. Furthermore, the incorporated hydrophilicity of the receptor beads caused the bilayer surface to be more open. The internal dendrimer beads (yellow) were modelled as hydrophobic. In addition the parameters defining the interaction between the hydrophobic lipid tails (cyan) and both types of dendrimer beads (red, yellow) were less repulsive than the parameters defining the interactions between the hydrophobic lipid tails (cyan) and the lipid headgroup beads (green, blue, orange, purple). As a result penetrability of the dendrimer increased from fp=0.02 to fp=0.1109 when interacting with the new bilayer with the more open surface (refer to Appendix 5). Solvent Bead (0) Surface Dendrimer Bead (+1) Internal Dendrimer Bead (0) Solvent Bead (0) 25 20 80 Surface Dendrimer Bead (+1) 20 25 28 Internal Dendrimer Bead (0) 80 28 25
  21. 21. 21 5.3 Simulation of the System with Dendrimer Heads modelled as Ligands Despite the non-steady-state morphological discrepancies and discussed deficiencies of the model, it was tested whether the generation of structures resembling those expected if clathrin-mediated endocytosis were responsible for transmembrane transport was possible. Further refinement of soft- core repulsion parameters was required. It was assumed that the surface amine functional groups of the dendrimer could substitute for ligand complexes (refer to Further Discussion). The effect of this assumption was tested by holding constant all soft-core repulsion parameters for the surface dendrimer beads excluding the parameter governing the surface dendrimer bead-receptor bead interaction (αRL). This parameter was, in many instances, modelled to be attractive in order to simulate the strong binding relationship between ligands and receptors.11 5.3.1 Initial Attempts to Generate Clathrin-mediated Endocytic Morphology The following structures were achieved after 105 timesteps (approximately 15 μs). The surface dendrimer bead soft-core interaction parameter was varied from a repulsive αRL=15 through to an attractive αRL=-30 (refer to Appendix 1: Table 7): Figure 18: Penetration profiles of simulations. For the purposes of clear illustration of the counterions and part of the bilayer structures have been omitted. In these simulations, encapsulation of the dendrimer was not observed. The asphericity and radii of gyration were larger than would be expected if encapsulation were responsible for penetration. Figure 19: Radii of gyration of simulated dendrimer- bilayer systems after approximately 105 timesteps. Figure 20: Asphericity of simulated dendrimer-bilayer systems after approximately 105 timesteps. αRL=15 αRL=0 αRL=-15 3.6 3.7 3.8 3.9 4 4.1 -20 -10 0 10 20 Radiusof GyrationRg(nm) Soft-core Repulsion (αRL) 0 0.01 0.02 0.03 0.04 0.05 0.06 -20 -10 0 10 20 Asphericity(As) Soft-core Repulsion (αRL)
  22. 22. 22 Further data is required if the observed trends of increasing radii of gyration and asphericity with the decrease in the soft-core interaction parameter αRL are to be substantiated. Due to computing constraints, an inadequate number of different values of αRL were tested. Furthermore, sampling at t=100,000 was not performed by calculating statistics from multiple simulation runs with the same αRL value. Only one simulation per tested value of αRL was able to be run and statistics were calculated from neighboring timesteps (98,000; 99,000; 100,000; 101,000; 102,000). As such, error bars of one standard deviation from the mean are depicted by these graphs. Repeated simulations would allow for the calculation of standard errors and greater confidence when interpreting results (refer to Suggestions for Further Work). Such trends would be expected in these non-steady-state snapshots as a reduction in αRL should allow the dendrimer structure to migrate through the bilayer more quickly. This is because the dendrimer heads are more strongly attracted to receptor beads on both sides of the membrane, stretching the dendrimer across the internal bilayer structure as encapsulation was not observed. Furthermore, the increasing attraction between the receptor heads and outer dendrimer beads led to a wider spread of dendrimer on the bilayer surfaces consistent with similar simulations without the receptor-simulating lipid inclusion.2 These expectations were further supported by penetration data. Penetration similarly appeared to increase with with a decrease in the soft-core interaction parameter αRL: Figure 21: Penetration of the dendrimer into the bilayer after approximately 105 timesteps. Again, due to variability and incompleteness of data, further simulation is required in order to test whether this observed and expected trend is statistically significant. This statistical insignificance of the data is further demonstrated by Figure 22 using confidence bands calculated to bound 95% of least-squares linear regressions that could be expected to be generated given the sampling methodology (refer to Appendix 6). Note that a linearly-increasing relationship could be plotted between αRL=-15 and αRL=15 within these confidence bands and that linearity is not expected necessarily. 0 0.05 0.1 0.15 0.2 0.25 -20 -10 0 10 20 Penetration(fp) Soft-core Repulsion (αRL)
  23. 23. 23 Figure 22: Penetration data bounded by linear regression confidence bands. In all simulations the dendrimers were able to penetrate the bilayer to varying degrees. In these simulations, the penetration was possible due to the hydrophobicity of the internal dendrimer beads (yellow) combined with the observation that the parameters defining the interaction between the hydrophobic lipid tails (cyan) and both types of dendrimer beads (red, yellow) (α=28) were less repulsive than the parameters defining the interactions between the hydrophobic lipid tails (cyan) and the lipid headgroup beads (green, blue, orange, purple) (α=40). As a result, the dendrimer did not simply adhere to the top surface of the bilayer. Instead the internal dendrimer beads bridged the structure as they were only weakly repelled by the internal hydrophobic section of the bilayer. The dendrimer head beads aggregated near the receptor beads on both sides of the bilayer due to the introduced receptor-ligand interaction. The structures generated were consistent with Yan and Yu’s findings that attraction between lipid tails and inner-dendrimer beads has a large effect on penetrability.2 When the magnitude of the attractive force was too great (αRL=-30), the self-assembling bilayer structure deteriorated (refer to Further Discussion). Figure 23: Deterioration of bilayer structure with αRL=-30 With αRL<-25, the attractive force between receptors and dendrimer heads was greater than the repulsive forces keeping like-beads apart from one another. This had the effect of creating physically unrealistic and unstable regions of high bead density. The chaos of the system was further
  24. 24. 24 compounded by these unstable regions crossing the boundaries of the simulation environment and transporting to the other side of the box. Subsequent simulations used values of αRL equal to and greater than -25. 5.3.2 Further Attempts to Generate Clathrin-mediated Endocytic Morphology In accordance with Yan and Yu’s previous findings, to inhibit the observed bridging the repulsive parameter governing the interaction between the internal dendrimer beads and the lipid tails was increased from αTU=28 to αTU=40 (refer to Appendix 1: Table 8).2 Various values of αRL were trialled from a repulsive αRL=15 through to an attractive αRL=-25. After longer simulations, 245 timesteps (approximately 36 μs), the following morphologies were observed: Figure 24: Penetration profiles of simulations. For the purposes of clear illustration of the counterions and part of the bilayer structures have been omitted. Very different structures were observed after this increase in αTU. For αRL=15 the repulsion was so great that the dendrimer did not even adhere to the surface of the bilayer. Hence, its profile is not shown. However, in the absence of direct interaction with the bilayer, the radius of gyration of the dendrimer was reduced significantly to 2.85 nm. Some axial stretching (As≈ 0.06) was caused by the migration of counterions which accumulated in the volume separating the dendrimer and bilayer. Nevertheless, this observation of significant shrinkage was welcome due to better agreement with empirical measurements and previously discussed concerns (refer to Figure 17). For αRL=0, the dendrimer adhered to the surface of the bilayer. Although not made clear by the cross-section exhibited in Figure 24, random forces permitted the jumping of some surface dendrimer beads, with internal dendrimer beads attached, across the bilayer as was even more strongly observed for the case where the receptor-ligand interaction was modelled as attractive with αRL=-15. Note for αRL=-15 this attractive force was large enough to cause lipids to migrate onto the dendrimer. For αRL=-25 this migration was pronounced. Additionally, the strength of the ligand-receptor attraction was much greater relative to the random forces and dendrimer beads could no longer jump across the membrane. Consequently, a morphology resembling endocytosis evolved. It was also observed that the ligand-simulating dendrimer beads became more locally concentrated as αRL decreased (refer to Further Discussion). αRL=0 αRL=-15 αRL=-25
  25. 25. 25 Figure 25: Radii of gyration of simulated dendrimer-bilayer systems after approximately 245 timesteps. As previously discussed, more simulations are required to make better supported assertions. The gyration radius was smallest for the dendrimer that did not bind to the bilayer (αRL=15). It increased with decreasing αRL as per the previous results generated (refer to Section 5.3.1) until it decreased with the onset of encapsulation for αRL=-25 (refer to Figure 25). Figure 26: Penetration of the dendrimer into the bilayer after approximately 245 timesteps. Penetrability statistics were consistent with the profiles generated as well as the negative correlation described in the analysis of earlier simulations (refer to Figure 26). Penetration was achieved by the lipids moving to encapsulate the dendrimer for αRL=-25 by direct penetration for αRL=0 and likely by a combination of both mechanisms for αRL=-15. Figure 27: Asphericity of simulated dendrimer-bilayer systems after approximately 245 timesteps. 2 2.5 3 3.5 4 -30 -20 -10 0 10 20RadiusofGyration Rg(nm) Soft-core Repulsion (αRL) 0 0.05 0.1 0.15 0.2 0.25 0.3 -30 -20 -10 0 10 20 Penetration(fp) Soft-core Repulsion (αRL) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 -30 -20 -10 0 10 20 Asphericity(As) Soft-core Repulsion (αRL)
  26. 26. 26 As discussed, the dendrimer that did not bind to the bilayer was axially stretched by the counterions. For αRL=0 most of the dendrimer was bound to the surface of the bilayer in a comparatively spherical shape. The asphericity increased for αRL=-15 as the dendrimer structure opened. Surprisingly, the structure most closely resembling encapsulation developed the most aspherical form. The large asphericity of the structure generated for αRL=-25 in Figure 24 can be explained by its morphological development depicted by Figure 27: Figure 28: Morphological progression for αRL=-25. Due to the strong attraction between the receptors and the ligand-simulating dendrimer beads the dendrimer initially flattened out over the surface of the bilayer. Subsequently, receptors migrated on top of the dendrimer. The flat shape of the dendrimer was maintained and after approximately 27 µs the dendrimer broke through to the other side of the bilayer due to a combination of random forces, receptor-ligand attraction and increased pressure exerted by the growing number of repulsive lipid tails on top of the dendrimer. Subsequently, more receptor beads migrated to cover this breach and structures resembling encapsulation were observed. It is believed that if the simulation were able to continue a steady-state would be reached with a more spherical dendrimer structure evolving and an approximately equal number of dendrimer beads concentrated on either side of the midline of the bilayer (fp≈0.5) (refer to Suggestions for Further Work). However, continued simulation was not possible due to computing constraints. The relationship between these observed morphologies to CME and other simulations will be discussed in Section 6. 0µs 9µs 18µs 27µs 36µs 45µs
  27. 27. 27 A phase table describing the morphologies for different soft-core repulsion parameters was generated: Table 3: Phase table describing different morphologies as a result of changing soft-core repulsion parameters. αRL αTU -25 -15 0 15 28 - Strong Direct Penetration Direct Penetration Direct Penetration 40 Endocytosis Direct Penetration With Pronounced Receptor Migration Adhesion with Direct Penetration No Adhesion Scope exists to further develop this table in conjunction with improving an understanding for how physical properties relate to the soft-repulsion parameters (refer to Suggestions Further Work).
  28. 28. 28 6.0 Further Discussion 6.1 Known Shortcomings of the Model Given the simplicity of the simulated membrane a true representation of the clathrin-mediated endocytosis was unlikely to have been simulated. Figure 29: Schematic illustrating more representative membrane complexity.43 This experiment used a tensionless membrane. In reality it has been shown that surface tension varies and that locally, negative surface tension can be generated by the cytoskeleton, dynamin or the actin filaments.39 Consequently, internalisation via invagination and cytotoxicity associated with hole generation (refer to Figure 30) were unlikely to be well-modelled by these simulations. Figure 30: Empirically observed hole inducing effects of cationic PAMAM dendrimers on tense lipid bilayer using atomic force microscopy; 44 see also 1, 36-38, 45, 46. Other simulations have regulated the tension of the bilayer. In order to maintain a constant surface tension, frequent transformations (refer to Equation 19) to the dimensions of the simulation box can be performed whilst keeping the simulation volume constant.2 (19)
  29. 29. 29 Alternatively, the N-varied DPD method maintains a constant membrane tension by adding and removing water and lipids from the edges of the simulation environment as required.39 Additionally, the receptor beads were too simple to form complex clathrin cages to encapsulate the dendrimer: Figure 31: A clathrin cage with a single triskelion (composed of three interlocking clathrin proteins) highlighted.47 In biological fluids, nanoparticles may interact with other circulated proteins and not with cellular membranes. Hence, simulating in a pure solvent environment may not be representative of reality even on such small length scales.7 The 1:3 ratio of lipid types was chosen hastily due to time pressures. Other studies have incorporated 1:1 ratio in accordance with current knowledge of mammalian membranes (refer to Suggestions for Further Work).13, 48 Also, the uniform distribution of receptors and ligand-simulating dendrimer beads on the surfaces of the membrane and dendrimer was not representative of a real system. It was assumed that the surface amine functional groups could substitute for ligands yielding this uniform distribution. Chemical synthesis of dendrimers is not able to achieve such uniformity. Synthesis of ligand coated PAMAM results in a distribution of the number of attached ligands. The number of attached ligands is much fewer in number than the number of surface amine groups.49 Simulating the effects of ligand distributions on cellular uptake has been studied but only using polygons as interacting particles (refer to Suggestions for Further Work).12 Consistent with Yan and Yu’s modelling, a number of counterions were included in the simulation and behaved according to the electrostatics detailed in Section 2.1 However, this implementation lacks refinement, further compounding potential sources of inaccuracy. Similar concerns stemming from the reuse of other parameters such as spring constants governing restorative forces exist. Finally, the energies associated with receptor-ligand adhesion are largely unknown.50 αRL is related to these unspecified energies. In order to observe endocytosis, αRL had to be made attractive enough to induce locally dense regions populated by receptor and ligand head. This modification did not disturb the thermostat; however, this clumping of beads may signify further deviation of the model from reality. Yang and Ma were incorporated a classical Lennard–Jones potential to govern the receptor-ligand interaction (refer to Figure 7) in order to prevent this issue.29
  30. 30. 30 6.2 Relationship of CME to Simulations Using this receptor model, encapsulation of the dendrimer was able to be simulated following changes to the soft-core repulsion parameters. This encapsulation occurred via morphologies varying from descriptions of CME and the hypothesis was consequently rejected. In the absence of invaginations and actin-induced regions of negative surface tension, internalisation of the vesicle was not observed via the ‘coated pit’ morphology depicted in Figure 5.51 Encapsulation was primarily observed to occur by the migration of receptors to the adhesive front.50 Consequently, encapsulation appeared to occur by a mechanism more closely resembling phagocytosis (refer to Figure 4). This mechanism is associated with larger length scales than that which was simulated.25 Further investigation is required to determine whether this model can be used for the purpose of simulating phagocytosis or revised to produce simulations that mimic CME morphological progression. Other simulations have been able to achieve similar structures resembling endocytosis. Macroscale physics have been incorporated into these mesoscale models in the form of Canham-Helfrich theory. Canham-Helfrich theory introduces membrane bending energies so as to limit direct penetration.39 It was possible to achieve this outcome without the need for this added complexity by balancing soft repulsion parameters. 6.3 The Notable Shortage of Empirical Data Experimentalists have not been able to accurately observe many proposed mechanisms for endocytosis due to technological limitations at these lengthscales and timescales (refer to Figure 32).52, 53 There is currently a pressing need to be able to effectively differentiate between pathways and perform quantitative analysis of transport mechanisms.54 Figure 32:Currently there is little overlap between what can experimentally be observed and what can be simulated when studying endocytosis.25 As mentioned previously, in order for the new model to be of value, it must reproduce solubilities, liquid compressibilities and the shape of interfaces on the length scale of the course-grained molecules.4 Such comparisons are currently limited by a deficiency in empirical data surrounding endocytic processes. As a consequence, simulations do little more than attempt to generate hypothesised morphological progression. Many attempts to incorporate the modest available data
  31. 31. 31 into the DPD model have been made or discussed (e.g. isolated dendrimer size, lipid diffusion, hole generation, hole size and lipid density). Due to the complexity of the system, many of the soft-core repulsion parameters use in this simulation were estimated by qualitative arguments coupled with iterative refinement or quoted from other studies which use similar methods.1, 17, 34 Some interaction parameters are able to be generated from Flory-Huggins theory and empirical testing.19 Calculating a complete set of Flory-Huggins 𝜒 parameters for complex systems, including those with multiple nanoscale species with Janus properties would be extremely difficult given the number of interaction pairs. Furthermore, energies associated with receptor-ligand binding, encapsulated gyration radii, internalisation velocities are all currently unknown. If the key issues concerning this model are addressed by reformulation, its future acceptance or rejection should be determined based on agreement with available endocytic data and practical value however blind (refer to Suggestions for Further Work).
  32. 32. 32 7.0 Conclusions Incorporation of a receptor-simulating lipid into the bilayer and subsequent tuning of soft-core repulsion parameters generated endocytic morphology. It was found that self-assembly of the bilayer structure was not compromised when converting a quarter of the simulated lipid molecules into a receptor-simulating lipid. These receptor-simulating lipids extended on average 0.4 nm further into the solvent than the DPPC simulating lipids. The lipid density of the bilayer increased by approximately 13% when compared with Yan and Yu’s previous simulations.2 The stability of the DPD model, on comparing with Yan and Yu’s previous simulations, was found to be compromised by changes made to the interaction parameters governing the generation of the G5 PAMAM dendrimer in solvent, causing it to be 40% more open than empirically observed (Rg=3.4 nm > Rg=2.4 nm). These changes were partially rectified for the simulations by reverting the interaction parameters to their historic values; however, these changes could not impose the required structural changes before interaction with the bilayer, increasing penetration. Despite these issues, (resulting in enhanced direct penetration,) further refinement of the soft-core repulsion parameters revealed that endocytosis could be modelled. Statistical analysis of changes to the soft-core interaction parameter governing the behaviour of proximal dendrimer surface beads and receptor beads suggested that penetrability could be increased if this repulsive force was instead modelled as attractive. The magnitude of this attraction was limited such that for αRL>-25 the bilayer structure became unstable due to the resultant creation of hyper-dense regions within the simulation box. The observed structures of dendrimers generated by these changes were open and distributed throughout the structure of the bilayer, again suggesting entry by direct penetration. Encapsulation of the dendrimer was promoted after subsequently increasing the repulsive force between the internal dendrimer beads and the lipid tails of the bilayer. From these results it was concluded that the binding energy between receptors and ligands as well as the hydrophobic interaction between the bilayer lipid tails and internal structure of the interacting particle has a large effect on binding and the type of penetration. Morphological changes in the simulation in which a form of endocytosis was observed may provide some insight as to how such encapsulation progresses. However, due to the discussed limitations of the model the morphology more closely matched a phagocytic mechanism. On the simulated length scale such a result was unexpected indicating that further refinement of the model may be needed (refer to Suggestions for Further Work) if CME, as it is currently understood, is to be simulated and the presently unobservable dynamics conjectured. As discussed, due to computational constraints, many of the results of the simulations run in this study were largely preliminary. Continued effort in this area of research will have future implications for drug development. When models can be used with confidence, molecular architectures will be able to be tested without costly synthesis. In order to generate robust models and build this confidence, continued interest in formulation and modification is required to better describe the intricacies of the very complex yet fundamental process of endocytosis.
  33. 33. 33 8.0 Suggestions for Further Work Future experiments require access to appropriate computing resources similar to the cluster systems housed at VLSCI. 8.1 Simple Immediate Revisions to the Current DPD Model As uncovered by the results generated, revisions to enhance the representativeness of the DPD model need to be made:  The assembly of the bilayer structure needs to resimulated with the ‘target lipid density’ set to 0.64 nm2 .  The prevalence of receptor-simulating lipids needs to be increased to approximately 50% in accordance with current knowledge of mammalian membranes.13, 48  The protrusion of these receptor lipids into the solvent may also need to be revised in light of information from empirical studies of bilayer structure.  Reversion to the original parameters governing the dendrimer-water system need to be readopted and simulated so that on introducing the dendrimer to the bilayer simulation box the structure of the dendrimer agrees with empirical measurements.  The introduction of an additional type of ligand bead distributed on the surface of the dendrimer in accordance with Mullen and Banaszak Holl’s findings needs to be incorporated to remove the erroneous assumption that surface amine functional groups can substitute for ligand complexes.49  To prevent the dense clumping of ligand and receptor beads, in accordance with Yang and Ma, this attractive pairwise interaction should be governed by a classical Lennard–Jones potential.29  The diffusion constant for membrane receptors in a bilayer is estimated to be approximately 104 nm2 /s.50 If accurate tracking of these receptors can be performed, αRL can be tuned to achieve expected diffusion.  The concentration of counterions and spring coefficients may require revision following examination of relevant literature. 8.2 Extension of Results Subsequent to the introduction of these changes, extended simulation is required to see whether CME can be observed. Multiple simulations running with the same set of soft-core interaction parameters are required so that sampling techniques and data interpretation can be strengthened. Simulations should ideally be run until steady-states have been reached so as to gain more a complete understanding of morphological progression. From these results comprehensive phase diagrams could be generated. It may be that CME still cannot be simulated following these simple revisions. A new model incorporating a more complex bilayer structure may be required. The distribution, size and structure of the receptors may need to be revised so that they are able to form clathrin cage-like configurations (refer to Figure 31). Similarly, strategies to control the surface tension of the bilayer on macro and local scales may need to be incorporated to simulate the roles of actin and dynamin in promoting invagination and vesicle release (refer to Figure 4).
  34. 34. 34 If CME is not observed following the simple revisions this model could still be extended to simulate a larger interacting particle instead of G5 PAMAM. It is known that phagocytosis occurs on larger length scales. Subsequent to further evaluation as to whether the endocytic morphology generated by this study resembles phagocytosis, it would be interesting to observe whether similar encapsulation could be induced without the need to incorporate energies associated with membrane deformation into the model, as is often done when simulating on larger length scales.
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  38. 38. 38 10.0 Appendices Appendix 1: Simulation Soft-core Repulsion Parameters Table 4: Historic soft-core repulsion parameters.1 Lipid Head (+1) Lipid Head (-1) Lipid Head (0) Lipid Tail (0) Solvent Bead (0) Surface Dendrimer Bead (+1) Counterions (-1) Internal Dendrimer Bead (0) Lipid Head (+1) 25 17 22 40 22 15 20 28 Lipid Head (-1) 17 25 22 40 22 15 20 28 Lipid Head (0) 22 22 25 40 25 15 22 28 Lipid Tail (0) 40 40 40 25 80 28 80 28 Solvent Bead (0) 22 22 25 80 25 20 20 80 Surface Dendrimer Bead (+1) 15 15 15 28 20 25 20 28 Counterions (-1) 20 20 22 80 20 20 25 28 Internal Dendrimer Bead (0) 28 28 28 28 80 28 28 25 Table 5: Soft-core repulsion parameters for bilayer self-assembly. Lipid Head (+1) Lipid Head (-1) Lipid Head (0) Lipid Tail (0) Solvent Bead (0) Receptor Head (0) Lipid Head (+1) 25 17 22 40 22 25 Lipid Head (-1) 17 25 22 40 22 25 Lipid Head (0) 22 22 25 40 25 25 Lipid Tail (0) 40 40 40 25 80 40 Solvent Bead (0) 22 22 25 80 25 15 Receptor Head (0) 25 25 25 40 15 25
  39. 39. 39 Table 6: Soft-core repulsion parameters for interacting bilayer-dendrimer system. Starred parameters were introduced not required by Yan and Yu’s receptorless system. These parameter were carefully chosen to model the receptor head similarly to other lipid heads in an attempt to reproduce Yan and Yu’s results.1, 2 Lipid Head (+1) Lipid Head (-1) Lipid Head (0) Lipid Tail (0) Solvent Bead (0) Receptor Head (0) Surface Dendrimer Bead (+1) Counterions (-1) Internal Dendrimer Bead (0) Lipid Head (+1) 25 17 22 40 22 25* 15 20 28 Lipid Head (-1) 17 25 22 40 22 25* 15 20 28 Lipid Head (0) 22 22 25 40 25 25* 15 22 28 Lipid Tail (0) 40 40 40 25 80 40* 28 80 28 Solvent Bead (0) 22 22 25 80 25 25* 15 20 80 Receptor Head (0) 25* 25* 25* 40* 25* 25* 15* 12* 28* Surface Dendrimer Bead (+1) 15 15 15 28 15 15* 25 20 28 Counterions (-1) 20 20 22 80 20 12* 20 25 28 Internal Dendrimer Bead (0) 28 28 28 28 80 28* 28 28 25 Table 7: Soft-core repulsion parameters for interacting bilayer and dendrimer. Lipid Head (+1) Lipid Head (-1) Lipid Head (0) Lipid Tail (0) Solvent Bead (0) Receptor Head (0) Surface Dendrimer Bead (+1) Counterions (-1) Internal Dendrimer Bead (0) Lipid Head (+1) 25 17 22 40 22 25 15 20 28 Lipid Head (-1) 17 25 22 40 22 25 15 20 28 Lipid Head (0) 22 22 25 40 25 25 15 22 28 Lipid Tail (0) 40 40 40 25 80 40 28 80 28 Solvent Bead (0) 22 22 25 80 25 25 15 20 80 Receptor Head (0) 25 25 25 40 25 25 αRL 12 28 Surface Dendrimer Bead (+1) 15 15 15 28 15 αRL 25 20 28 Counterions (-1) 20 20 22 80 20 12 20 25 28 Internal Dendrimer Bead (0) 28 28 28 28 80 28 28 28 25
  40. 40. 40 Table 8: Revised soft-core repulsion parameters for interacting bilayer and dendrimer. Lipid Head (+1) Lipid Head (-1) Lipid Head (0) Lipid Tail (0) Solvent Bead (0) Receptor Head (0) Surface Dendrimer Bead (+1) Counterions (-1) Internal Dendrimer Bead (0) Lipid Head (+1) 25 17 22 40 22 25 15 20 28 Lipid Head (-1) 17 25 22 40 22 25 15 20 28 Lipid Head (0) 22 22 25 40 25 25 15 22 28 Lipid Tail (0) 40 40 40 25 80 40 28 80 αTU Solvent Bead (0) 22 22 25 80 25 25 15 20 80 Receptor Head (0) 25 25 25 40 25 25 αRL 12 28 Surface Dendrimer Bead (+1) 15 15 15 28 15 αRL 25 20 28 Counterions (-1) 20 20 22 80 20 12 20 25 28 Internal Dendrimer Bead (0) 28 28 28 αTU 80 28 28 28 25
  41. 41. 41 Appendix 2: MATLAB Code for Bilayer Bead Density Profile % Bilayer Bead Density by Displacement from Average z-coordinate %create matrix with data from binary output file [file,path]=uigetfile('SpaGe1.bin','Open the file');%to save N(r)_r data if file fid=fopen([path,file],'r'); a=fread(fid,[4 inf],'float32'); end %calculate zavg ztot=0; for i=1:numel(a)/4; z = a(3,i); ztot = ztot+z; end zavg= ztot/(numel(a)/4); %avg displacement from zavg density= zeros(2,numel(a)/4); for i=1:numel(a)/4; density(1,i)=(a(3,i)-zavg); density(2,i)=a(4,i); end %partition 0.05nm x=linspace(-19.975, 20.025, 800); for j=1:numel(a)/4 for i=1:numel(x) if density(1,j)<= (x(i)+0.025) && density(1,j)>(x(i)-0.025) density(1,j)=x(i); end end end %displacement histogram for relevant bilayer beads (types 1, 2, 3, 4, 6) y=zeros(6,numel(x)); for i=1:6 for j=1:numel(a)/4 for m=1:numel(x) if density(2,j)== i && density(1,j)== x(m) y(i,m)=y(i,m)+1/88.2; %volume of xy-sliver 88.2nm end end end end y=[y(1,:);y(2,:);y(3,:);y(4,:);y(6,:)];
  42. 42. 42 y=y'; %histogram smoothing smooth = sgolayfilt(y,3,29); subplot(2,1,1) plot(x, smooth,'linewidth', 2) axis([-5 5 0 3.5]) title('Displacement Distribution of Bilayer Beads from Central z- coordinate') xlabel('Displacement from Bilayer Centre (nm)'),ylabel('Number Density (beads/nm^3)'); legend('Lipid Head (+1)','Lipid Head (-1)','Lipid Head (0)','Lipid Tail (0)','Receptor Head (0)') subplot(2,1,2) plot(x,smooth, 'linewidth', 2) axis([0 5 0 1]) title('Magnified Displacement Distribution of Bilayer Beads from Central z- coordinate') xlabel('Displacement from Bilayer Centre (nm)'),ylabel('Number Density (beads/nm^3)'); legend('Lipid Head (+1)','Lipid Head (-1)','Lipid Head (0)','Lipid Tail (0)','Receptor Head (0)')
  43. 43. 43 Appendix 3: Asphericity Asphericity of the dendrimer can be calculated from the following equations: (20) Where I1 and I2 are the first two invariants: (21) (22) 𝜆1, 𝜆2, 𝜆3 are the eigenvalues of the gyration tensor: 𝑮 𝑻 = 𝟏 𝑵 ∑ (𝒔𝒊⨂𝒔𝒊)𝑵 𝒊=𝟎 (23) 𝒔𝒊 = 𝒓𝒊 − 𝒓 𝑪𝑴 (24) % Asphericity MATLAB %create matrix with data from binary output file [file,path]=uigetfile('SpaGe1.bin','Open the file');%to save N(r)_r data if file fid=fopen([path,file],'r'); a=fread(fid,[4 inf],'float32'); end %remove counterions b=a; count=0; for i=1:numel(a)/4; if a(4,i)== 8 b(:, i-count) = []; count=count+1; else b(:,i-count)= a(:,i); end end %calculate centre of mass (all beads have the same mass) xtot=0; ytot=0; ztot=0; for i=1:numel(b)/4; x = b(1,i); xtot = xtot+x; end xavg= xtot/(numel(b)/4); for i=1:numel(b)/4; y = b(2,i);
  44. 44. 44 ytot = ytot+y; end yavg= ytot/(numel(b)/4); for i=1:numel(b)/4; z = b(3,i); ztot = ztot+z; end zavg= ztot/(numel(b)/4); rcm=[xavg;yavg;zavg] %gyration tensor dispoutertot=zeros(3); disp = [0;0;0]; for i=1:numel(b)/4; for j=1:3 disp(j,1)=b(j,i)-rcm(j,1); end dispouter=disp*disp'; dispoutertot=dispoutertot+dispouter; end Gt=dispoutertot/(numel(b)/4) %calculate asphericity eigvals=eig(Gt); I1= sum(eigvals); I2= eigvals(1)*eigvals(2)+eigvals(2)*eigvals(3)+eigvals(3)*eigvals(1); As=1-(3*(I2)/(I1)^2)
  45. 45. 45 Appendix 4: Radius of Gyration The radius of gyration is determined by the following equation: (25) Where the summation runs over all N beads in the dendrimer at positions ri. rcm denotes the center of mass of the dendrimer. % Gyration Radius MATLAB %create matrix with data from binary output file [file,path]=uigetfile('SpaGe1.bin','Open the file');%to save N(r)_r data if file fid=fopen([path,file],'r'); a=fread(fid,[4 inf],'float32'); end %remove counterions b=a; count=0; for i=1:numel(a)/4; if a(4,i)== 8 b(:, i-count) = []; count=count+1; else b(:,i-count)= a(:,i); end end %calculate centre of mass (all beads have the same mass) xtot=0; ytot=0; ztot=0; for i=1:numel(b)/4; x = b(1,i); xtot = xtot+x; end xavg= xtot/(numel(b)/4); for i=1:numel(b)/4; y = b(2,i); ytot = ytot+y; end yavg= ytot/(numel(b)/4); for i=1:numel(b)/4; z = b(3,i); ztot = ztot+z; end zavg= ztot/(numel(b)/4); rcm=[xavg;yavg;zavg]
  46. 46. 46 %squared displacement from centre of gravity squared dispsqutot=[0;0;0]; disp = [0;0;0]; for i=1:numel(b)/4; for j=1:3 disp(j,1)=b(j,i)-rcm(j,1); end dispsqu = disp.^2; dispsqutot=dispsqutot+dispsqu; end sum(dispsqutot)/((numel(b)/4)); Rg = sqrt(sum(dispsqutot)/((numel(b)/4)))
  47. 47. 47 Appendix 5: Penetrability Penetrability was defined as the fraction of dendrimers that were below the average z-coordinate of all lipid beads enclosed by the maximum and minimum x,y coordinates of the dendrimer beads at the fringes of the dendrimer structure. % Penetrability MATLAB %create matrix with data from binary output file [file,path]=uigetfile('SpaGe1.bin','Open the file');%to save N(r)_r data if file fid=fopen([path,file],'r'); a=fread(fid,[4 inf],'float32'); end [file,path]=uigetfile('SpaGe2.bin','Open the file');%to save N(r)_r data if file fid=fopen([path,file],'r'); c=fread(fid,[4 inf],'float32'); end %remove counterions b=c; count=0; for i=1:numel(c)/4; if c(4,i)== 8 b(:, i-count) = []; count=count+1; else b(:,i-count)= c(:,i); end end %box coordinates to determine relevant bilayer beads for penetrability dimmax=max(b,[],2); dimmin=min(b,[],2); dimensions=[dimmin(1,1),dimmin(2,1); dimmax(1,1),dimmax(2,1)]; %remove irrelevant bilayer beads d=a; count=0; for i=1:numel(a)/4; if a(1,i)<=dimensions(1,1)|a(2,i)<=dimensions(1,2)|a(1,i)>=dimensions(2,1)|a(2 ,i)>=dimensions(2,2) d(:, i-count) = []; count=count+1; else d(:,i-count)= a(:,i); end end %calculate average z coordinate of remaining bilayer (all beads have the same mass) ztot=0; for i=1:numel(d)/4;
  48. 48. 48 z = (3,i); ztot = ztot+z; end zavg= ztot/(numel(d)/4) test=ones(1,numel(b)/4); %test if dendrimer below middle of bilayer for i=1:numel(b)/4 if b(3,i)<=zavg test(1,i)=1; else test(1,i)=0; end end %fraction below middle of bilayer fp=sum(test)/numel(test)
  49. 49. 49 Appendix 6: Confidence Bands Data file: aRL fp 15 0.1109 15 0.1327 15 0.1426 15 0.1485 15 0.1089 0 0.1386 0 0.1762 0 0.1762 0 0.1802 0 0.2 -15 0.2693 -15 0.2178 -15 0.2495 -15 0.1881 -15 0.1861 R Code: data = read.delim(file.choose(),header=T) X=data$aRL Y=data$fp plot(X, Y, xlab="aRL", ylab="Penetrability fp", xlim=c(-20, 20), ylim=c(0, 0.4)) ml=lm(Y~X) %abline(ml) summary(ml) confint(ml,level=0.95) bands=predict(ml,newdata=data.frame(X),interval=c("prediction" ),level=0.95) lines(X, bands[, "lwr"], lty = "dotted") lines(X, bands[, "upr"], lty = "dotted")
  50. 50. 50 Appendix 7: Intermediate Data Table 9: Rg data for αTU=28 αRLTimestep 98000 99000 100000 101000 102000 μ σ 15 3.8E+00 3.7E+00 3.7E+00 3.7E+00 3.7E+00 3.7E+00 2.5E-02 0 3.9E+00 3.8E+00 3.9E+00 3.8E+00 3.9E+00 3.9E+00 5.2E-02 -15 4.0E+00 4.1E+00 4.0E+00 4.0E+00 4.0E+00 4.0E+00 3.1E-02 Table 10: As data for αTU=28 αRLTimestep 98000 99000 100000 101000 102000 μ σ 15 3.3E-02 1.9E-02 5.3E-03 1.1E-02 1.3E-02 1.6E-02 1.1E-02 0 3.8E-02 3.4E-02 3.2E-02 1.5E-02 2.7E-02 2.9E-02 9.1E-03 -15 2.4E-02 4.3E-02 3.1E-02 3.3E-02 5.6E-02 3.7E-02 1.3E-02 Table 11: fp data for αTU=28 αRLTimestep 98000 99000 100000 101000 102000 μ σ 15 1.1E-01 1.3E-01 1.4E-01 1.5E-01 1.1E-01 1.3E-01 1.8E-02 0 1.4E-01 1.8E-01 1.8E-01 1.8E-01 2.0E-01 1.7E-01 2.2E-02 -15 2.7E-01 2.2E-01 2.5E-01 1.9E-01 1.9E-01 2.2E-01 3.7E-02 Table 12: Rg data for αTU=40 αRLTimestep 238000 239000 240000 241000 242000 μ σ 0 3.2E+00 3.2E+00 3.2E+00 3.3E+00 3.2E+00 3.2E+00 4.2E-02 15 2.8E+00 2.8E+00 2.8E+00 2.9E+00 2.9E+00 2.9E+00 7.1E-02 -15 3.7E+00 3.7E+00 3.5E+00 3.6E+00 3.6E+00 3.6E+00 7.9E-02 -25 3.3E+00 3.3E+00 3.3E+00 3.3E+00 3.3E+00 3.3E+00 3.1E-02 Table 13: As data for αTU=40 αRLTimestep 238000 239000 240000 241000 242000 μ σ 0 3.2E-02 4.6E-02 3.6E-02 2.7E-02 2.6E-02 3.3E-02 8.5E-03 15 4.4E-02 3.8E-02 5.1E-02 1.0E-01 8.6E-02 6.4E-02 2.8E-02 -15 5.6E-02 7.6E-02 5.2E-02 3.5E-02 4.5E-02 5.3E-02 1.5E-02 -25 1.2E-01 1.3E-01 1.3E-01 1.2E-01 1.1E-01 1.2E-01 8.4E-03 Table 14: Rg data for αTU=40 αRLTimestep 238000 239000 240000 241000 242000 μ σ 0 1.1E-01 1.1E-01 1.1E-01 1.1E-01 8.5E-02 1.1E-01 1.2E-02 15 0.0E+00 0.0E+00 0.0E+00 0.0E+00 0.0E+00 0.0E+00 0.0E+00 -15 1.5E-01 1.7E-01 1.6E-01 1.3E-01 1.6E-01 1.5E-01 1.2E-02 -25 2.7E-01 2.6E-01 2.3E-01 2.6E-01 2.7E-01 2.6E-01 1.8E-02

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