Molecular Dynamics Studies of PEGylated α-Helical Coiled Coils and

Jul 7, 2014 - Handan Acar , Samanvaya Srivastava , Eun Ji Chung , Mathew R. Schnorenberg , John C. Barrett , James L. LaBelle , Matthew Tirrell...
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Molecular Dynamics Studies of PEGylated α‑Helical Coiled Coils and Their Self-Assembled Micelles Sun Young Woo and Hwankyu Lee* Department of Chemical Engineering, Dankook University, Yongin 448-701, South Korea ABSTRACT: We performed coarse-grained (CG) molecular dynamics simulations of trimeric α-helical coiled coils grafted with poly(ethylene glycol) (PEG) of different sizes and conjugate positions and the self-assembled micelle of amphiphilic trimers. The CG model for the trimeric coiled coil is verified by comparing the α-helical structure and interhelical distance with those calculated from all-atom simulations. In CG simulations of PEGylated trimers, the end-to-end distances and radii of gyration of PEGs grafted to the sides of peptides become shorter than those of free PEGs in water, which agrees with experiments. This shorter size of the grafted PEGs is also confirmed by calculating the thickness of the PEG layer, which is less than the size of the mushroom. These indicate the adsorption of PEG chains onto coiled coils since hydrophobic residues in the exterior sites of coiled coils tend to be less exposed to water and thus interact with PEGs, leading to the compact conformation of adsorbed PEGs. Simulations of the self-assembly of amphiphilic trimers show that the randomly distributed trimers self-assemble to micelles. The outer radius and hydrodynamic radius of the micelle, which were calculated respectively from radial densities and diffusion coefficients, are ∼7 nm, in agreement with the experimental value of ∼7.5 nm, while the aggregation number of amphiphilic molecules per micelle is lower than the experimentally proposed value. These simulations predict the experimentally measured size of PEGs grafted to the trimeric coiled coils and their self-assembled amphiphilic micelles and suggest that the aggregation number of the micelle may be lower, which needs to be confirmed by experiments.

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coils and PEG conjugation.16−18 The Kros group found the effects of PEGylation on the structural stability of coiled coils and coiled-coil-induced membrane fusion.19−23 Recently, the Xu group attached PEG chains to the three-stranded α-helical coiled coils, and their scattering experiments showed the effect of the PEG size and conjugation architecture (side vs end conjugation) on the size and structural stability of coiled coils and their self-assembled micelles.24−28 However, the structure factors from the scattering data were interpreted with the simple modeling of assuming coiled-coil bundles to be parallel cylinders because of the limited resolution, and hence the conformation and structure of the PEGylated coiled coils and the self-assembled micelle need to be studied at nearly the atomic scale. Molecular dynamics (MD) simulations have been performed to understand the structural stability and conformational transition of α-helical, β-stranded, or cyclic peptides grafted with PEGs.29−31 For α-helical coiled coils, simulation studies have mostly focused on the structural stability modulated by individual amino acids.32−35 For the coiled-coil−PEG complex, Jain and Ashbaugh performed replica exchange MD simulations of a single PEGylated coiled coil at different temperatures,

INTRODUCTION Amphiphilic peptide−polymer conjugates have shown great potential for biomedical applications since the combination of peptides and polymers can increase the structural stability, solubility, biocompatibility, and circulating lifetime in blood.1−5 Their self-assembled structures, functionalities, and thermodynamic properties can be easily controlled by conjugating different amino acids and synthetic polymers, and hence they have been mainly used as stimuli-responsive transporters that can deliver genes or drug molecules to the desired site.2,6−8 In particular, coiled coils, which consist of two or more α-helices wound into a superhelix, have been widely used for the peptide−polymer conjugate since they can easily self-assemble to mechanically rigid structures,9−12 which are induced by the hydrophobic interactions in the core of the coiled coil and stabilized by the electrostatic interactions between charged residues. To improve the solubility, structural stability, and circulating lifetime, poly(ethylene glycol) (PEG) has been attached to the self-assembled coiled coils in a process called PEGylation.13−15 Experimentally, the Tirrell group pioneered the synthesis of block copolymers composed of coiled coils and PEGs, showing that self-assembled structures can be controlled by different temperatures and pH.2 The Klok group also synthesized the PEG coiled-coil copolymers and characterized their structures and dynamics, which are modulated by different types of coiled © 2014 American Chemical Society

Received: May 21, 2014 Revised: June 28, 2014 Published: July 7, 2014 8848

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Figure 1. (a) Structure and helical wheel diagram of the trimeric coiled coil (1CW). Peptides are represented as ribbons with the hydrophobic core highlighted as gray dots. For the wheel diagram, amino acid sequences are plotted clockwise. (b) Structures of PEGylated trimers for the side conjugation (top) and the end conjugation (bottom). (c) Structures of the amphiphilic trimer and its monomer. For peptides, blue, gray, and green colors, respectively, represent charged, hydrophobic, and hydrophilic amino acids. Light-blue and red dots represent the alkyl chains (C18) and PEG chains, respectively. The images were created with Visual Molecular Dynamics.60 developed respectively by Marrink et al.44,45 and our group.46−48 The structure and coordinates of designed peptide 1CW (EVEAL EKKVA ALESK VQALE KKVEA LEHG), which is known to form the trimeric coiled coil, were downloaded from the Protein Data Bank (PDB code 1COI).49 The 14th residue (serine (Ser)) of the peptide was replaced by cysteine (Cys), and N- and C-terminal groups were respectively acetylated (unprotonated) and amidated (protonated) to make them electrostatically neutral, matching the amino-acid sequence and the terminal charge state in the experiment.24 A temperature of 298 K and a pressure of 1 bar were maintained by applying the velocity-rescale thermostat50 and Berendsen barostat in the NPT ensemble.51 The LINCS algorithm was used to constrain the bond lengths.52 Simulations were performed at computational facilities supported by the National Institute of Supercomputing and Networking/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2013-C2-46). All-Atom Simulations of Peptides in Water. Three α-helices were aligned in parallel, with the position of hydrophobic residues in the core and the distance between the centers of mass (COM) of αhelices initially set to 1 nm, which is close to the conformation of typical three-stranded coiled coils (Figure 1a). This three-helix bundle was solvated with ∼10 000 water molecules in a periodic box of size 7 nm/side, and 6 counterions (Na+) were added to neutralize the system. A real-space cutoff of 11 Å (OPLS) or 14 Å (GROMOS96) was used for the Lennard-Jones (LJ) and electrostatic interactions with the inclusion of the particle mesh Ewald summation53 for long-range electrostatics. Simulations were performed for 100 ns with a time step of 2 fs, and the last 50 ns was used for analyses. CG Simulations of PEGylated Peptides in Water. Three copies of the CG α-helix were initially bundled in the same orientation and interhelical distance as described above for all-atom simulations. The bundle was equilibrated in water for 100 ns and verified by comparing with all-atom simulations, which will be described in the Results and Discussion section. This equilibrated three-helix bundle was used to generate the PEGylated trimer. The terminal bead of PEG (Mw = 1000, 2000, 5000, and 7000) was linked to the terminal bead in the side chain of Cys (14th residue) by applying the bond and angle potentials used for the CG PEG model, leading to the three-helix

showing that larger PEG chains stabilize the helical structure of coiled coils because of the interaction between PEGs and lysine residues of the peptide.36 Recently, Hamed et al. also found a higher helicity and lower solvent-accessible surface area for coiled coils grafted with larger PEGs because PEG chains interact with cationic lysines of coiled coils as well as with hydrophobic residues of coiled coils.37 Although these simulations have revealed the atomic-scale interactions between PEGs and coiled coils, the effects of PEG size and conjugate architecture on the size and self-assembled structure of the PEGylated coiled coil have not yet been computationally studied. In this work, we therefore perform MD simulations of PEGylated 3-helical coiled coils (trimers) and their selfassembled micelles. First, the secondary structure and conformation of the trimeric coiled coil are compared in allatom and coarse-grained (CG) simulations, which verify the CG model for the trimer. Second, we perform CG simulations of PEGylated trimers with different PEG lengths and grafting positions. The sizes of the grafted PEGs and free PEGs are compared to those measured from experiments, showing the decreased size of the grafted PEGs, which is rationalized by the hydrophobic interaction between PEGs and coiled coils. Finally, CG simulations of multiple copies of amphiphilic trimers, which are PEGylated trimers modified with alkyl chains and additional PEGs, show the self-assembly of micelles. The size and aggregation number of micelles are analyzed, which also compare with experimental observations.

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METHODS

All simulations and analyses were performed using the GROMACS4.5.5 simulation package.38−40 For all-atom simulations, the GROMOS96-54a741 and OPLS42,43 all-atom force fields (FFs) were used respectively with SPC and TIP4P water models, while CG models were taken from the MARTINI protein and PEG FFs 8849

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bundle grafted with three PEG chains (Figure 1b). This PEGylated trimer was solvated with 7000−44 000 CG waters in a periodic box of size 10−20 nm/side. In addition to the conjugation of PEG to the side of the peptides, PEG chains (Mw = 1000 and 2000) were also linked to the N-terminal bead (glutamic acid) of two peptides and to the Cterminal bead (glycine) of the other peptide (Figure 1b), which were solvated with 14 000−21 000 CG waters and 6 counterions (Na+) in a periodic box of size 12−14 nm/side. A cutoff of 12 Å was applied for LJ potential and electrostatic forces. The LJ and Coulomb potentials were smoothly shifted to zero between 9 and 12 Å and between 0 and 12 Å, respectively. Simulations were carried out for 0.8−1 μs with a time step of 8 fs. CG Simulations of Micelles in Water. To simulate the selfassembled micelle of a PEGylated trimer, the experimentally designed amphiphilic subunits were generated by attaching alkyl chains (C18) and an additional PEG to the PEGylated trimer (Figure 1c). The alkyl chain was obtained from the MARTINI model of the distearoylglycerophosphocholine (DSPC) lipid by removing the DSPC headgroup and replacing glycerol beads with hydrophilic CG bead P4. Then, the P4 bead of this alkyl chain was attached to the N-terminal bead of the peptide. A lysine residue was added to the C terminus of the peptide and linked to an additional PEG chain (Mw = 750), leading to the amphiphilic subunit synthesized in the experiment.26 Several copies (30, 40, 50, and 60) of amphiphilic trimers were randomly distributed in a periodic box of size 20 nm/side and solvated with ∼50 000 CG water molecules and 180−360 counterions (Na+). Simulations were performed for 1.5 μs with a time step of 8 fs, and the last 900 ns was used for analysis.

indicates the self-assembly of 50 amphiphilic trimers (150 amphiphilic molecules) with PEG of Mw = 2000 grafted to the side of coiled coils. Since experiments showed that the end conjugation of PEGs of Mw > 2000 destabilizes the trimeric αhelical structure,28 PEGs with Mw of 1000 and 2000 were simulated for the systems with the end PEGylation, while PEGs with Mw of 1000, 2000, 5000, and 7000 were simulated for the systems with side PEGylation. Secondary Structure and Conformation of the Trimeric Coiled Coil. Experimentally, peptide 1CW has been observed to form a trimeric coiled coil.49 To test the ability of different FFs to predict the structure and conformation of the trimeric coiled coil, the trimer was simulated in water with the OPLS and GROMOS96-54a7 allatom FFs and the MARTINI CG FF. The secondary structure of the trimer was calculated using the DSSP program,54 showing the helicity of more than 90% for both OPLS and GROMOS96-54a7 FFs, which indicates the formation of the stable coiled coil. Note that in the CG model the helical structure is fixed and thus does not change for the whole simulation time. To investigate the conformation of the trimer, we calculated the interhelical distance, which is defined as the distance between the COM of the backbone atoms for the 1st−4th, 13th−16th, or 26th−29th residues of each helix. Table 2 shows the interhelical distance of ∼1 nm for both all-atom and CG models, which is close to the interhelical distance of typical coiled coils in the experiment and theoretical model.55 These indicate that both all-atom and CG models can predict the secondary structure and conformation of the trimeric coiled coil and also confirm that when helices are simply modeled as parallel cylinders the theoretical assumption of a cylindrical radius of 5 Å is reasonable.28 Conformation of PEGs Grafted to the Trimeric Coiled Coil. The trimeric coiled coils were modified by conjugating PEG chains to either the side or end of each helix and simulated for 0.8−1 μs. Figure 2 shows the initial and final configurations of CG simulations with the PEGylated trimer. Starting with the initial PEG position far from the peptide, PEG chains become close to the peptides and wrap them, indicating the interaction between PEG chains and peptides. The extent of wrapping the peptide is greater for larger PEGs than for smaller PEGs. In Figure 3, the root-mean-squared end-to-end distances (⟨h2⟩1/2) and radii of gyration (Rg) averaged for three PEGs of each trimer are shown as a function of time. Both values reach steady states within 300 ns, indicating the equilibrated conformation of the grafted PEGs, although those for large PEGs (Mw = 7000) fluctuate greatly. In Figure 3, the ⟨h2⟩1/2 and Rg values for three individual PEG chains in T1-P5S do not differ significantly, which confirms that 1/2 and Rg can be reasonably averaged over three PEG chains. Figure 4 compares the average ⟨h2⟩1/2 and Rg of free PEGs with those of PEGs grafted to the side or end of the trimer. Note that free PEG chains (Mw = 550−7000) in water were previously simulated with the CG PEG FF developed by our group, and ⟨h2⟩1/2 and Rg agree well with experiments and polymer theories.46 Here, ⟨h2⟩1/2 and Rg from simulations are slightly shorter than those measured from experiments, apparently because experiments were performed with the maleimide-conjugated PEGs that should be larger than pure PEGs. For PEGs with a Mw of 1000 to 5000, ⟨h2⟩1/2 and Rg of the side-conjugated PEGs are respectively 22−25 and 7−16% shorter than those of free PEGs, which indicates the decreased

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RESULTS AND DISCUSSION PEGylated trimeric coiled coils and their self-assembled micelles were simulated with different PEG sizes and conjugate positions. Simulated systems are listed in Table 1, where T and Table 1. List of Simulations name allatom CG

no. of the trimer

PEG Mw

simulation time (μs)

1CW

T1

1

1CW side-PEGylated 1CW

T1 T1-P1S T1-P2S T1-P5S T1-P7S T1-P1E T1-P2E T30P2SC18 T40P2SC18 T50P2SC18 T60P2SC18

1 1 1 1 1 1 1 30

1000 2000 5000 7000 1000 2000 2000

0.1 0.8 0.8 0.8 1 0.8 0.8 1.5

40

2000

1.5

50

2000

1.5

60

2000

1.5

end-PEGylated 1CW amphiphilic subunit (micelle simulation)

0.1

the first number indicate the number of coiled-coil trimer. For the systems with PEGs, P and the following number designate the PEG’s molecular weight (P1, P2, P5, and P7 for PEG1000, PEG2000, PEG5000, and PEG7000), which are followed by S (side PEGylation) or E (end PEGylation). The last letter C18 indicates the alkyl-chain-attached amphiphilic subunit. For example, T1-P1E designates a single trimer with PEG of Mw = 1000 grafted to the end of coiled coils, and T50-P2S-C18 8850

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Table 2. Interhelical Distances between Centers of Mass (COM) for the Backbone Atoms of Four Residues in Three Different Positions (N- and C-Terminals and the Middle) of Each Helixa simulation all-atom

a

backbone atoms

OPLS

GROMOS96-54a7

coarse-grained

theoretical model28,55

1Glu-4Ala 13Glu-16Val 26Leu-29Gly

1.024 ± 0.007 1.053 ± 0.009 1.114 ± 0.159

1.056 ± 0.003 1.078 ± 0.002 0.853 ± 0.058

1.159 ± 0.023 1.033 ± 0.014 0.870 ± 0.103

1 1

All quantities are in nanometers.

Figure 2. Snapshots at the beginning (0 μs, first column) and end (0.8 or 1 μs, second−sixth columns) of simulations of the PEGylated trimer. The initial configuration is shown only for T1-P2S, but for other systems the grafted PEGs are also initially positioned far from the peptide. For clarity, water and ion molecules are omitted.

Figure 3. Root mean squared end-to-end distances (1/2) and radii of gyration (Rg) averaged for three PEGs grafted to the side of the trimeric coiled coil (left), and 1/2 and Rg for individual PEGs in T1-P5S (right), as a function of time.

size of the grafted PEGs, presumably because of the interaction between peptides and PEGs. Although the side-conjugated PEGs become more compact, the end-conjugated PEGs are almost the same size as free PEGs, indicating no conformational transition of the end-conjugated PEGs, in agreement with Lund et al.’s experiments.28 For PEG7000, ⟨h2⟩1/2 and Rg are close to those for free PEGs, indicating that the conformation of such a large PEG is not significantly influenced by the peptide−PEG interaction. Interactions between PEGs and Coiled Coils. As discussed above, the PEGylation on the side of coiled coils induces a more compact conformation of PEGs, which may be influenced by the interaction between PEGs and peptides. To understand this, the conformation of PEGs grafted to the side of peptides was analyzed. Figure 5 shows radial density profiles of PEGs (number of PEG beads) as a function of distance from the peptide. Larger PEGs form a thicker PEG layer on the peptide, as expected. These density profiles were used to calculate the thickness of the PEG layer on the trimer. Although ⟨h2⟩1/2 of the grafted PEGs and the thickness of the PEG layer are not necessarily exactly equal, they are likely to be close. In Figure 5, ⟨h2⟩1/2 values of PEGs are in the density range of 90−

Figure 4. Average ⟨h2⟩1/2 and Rg of PEGs grafted to the side or end of the coiled coil or those isolated in water. The experimental values of Rg are obtained from ref 28.

95%, and thus the range for the thickness of the PEG layer was determined by calculating 90−95% of the density of PEGs on the trimer and then comparing with the Flory radius, RF = aN3/5, where N is the degree of polymerization and a is the monomer size (3.3 Å for the bond length in CG model).56 The Alexander−de Gennes theory describes that at low grafting 8851

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(Leu), and valine (Val) show the lowest peaks, since those residues are mostly located in the interior core of the trimeric coiled coil and thus are sterically shielded (Figure 1), where PEGs cannot easily interact with those residues. This trend is similarly observed for systems with other PEG sizes and end conjugation, again confirming strong hydrophobic interactions. This indicates that PEG chains interact with coiled coils, since hydrophobic Ala residues, which are located in the exterior site of coiled coils, tend to be less exposed to water and thus interact with PEGs rather than with water, as also observed from Hamed et al.’s all-atom simulations.37 Self-Assembly of PEGylated Amphiphilic Micelles. Experimentally, Dong et al. conjugated two C18 alkyl chains and an additional PEG (Mw = 750) to the PEGylated trimer (the side-conjugated PEG of Mw = 2000) and found that those amphiphilic molecules self-assemble to micelles, showing the high efficiency of drug delivery.26 They also characterized the micelle size (hydrodynamic diameter) of ∼15 nm and proposed that the aggregation number of amphiphilic molecules is ∼45 per micelle (corresponding to ∼15 trimers).27 To confirm the experimentally measured size and proposed aggregation number, we simulated self-assemblies of amphiphilic trimers of 30, 40, 50, and 60, which respectively correspond to amphiphilic molecules of 90, 120, 150, and 180. Figure 7 shows

Figure 5. Number of PEG beads as a function of distance from the peptide (top panel) and thickness of the PEG layer, calculated from 90−95% densities, as a function of the grafted PEG molecular weight (bottom panel). A dotted line represents the size of the mushroom (RF) calculated from the Alexander−de Gennes theory.56

density the polymer chain on the planar nonadsorbing surface behaves like an isolated chain in solution, leading to a hemisphere (mushroom) conformation with a size given by RF.56 Note that the trimer surface is not planar, thus the discrepancy between simulation results and theory may occur. Figure 5 shows that for PEG1000, PEG2000, and PEG5000, the RF values are larger than the thickness of the PEG layer, indicating that PEGs grafted onto the trimer are more compact than free PEG chains in water, which implies the strong adsorption of PEGs onto the coiled coil. For PEG7000, the thickness of the PEG layer corresponds to RF, presumably because PEG7000 is too long to be influenced by the interaction with coiled coils, which is consistent with Figure 4 that also shows no difference in the PEG size between free PEG7000 and the side-conjugated PEG7000. Figure 6 shows radial distribution functions (RDFs) between PEGs and individual amino acids for T1-P1S. Alanine (Ala) shows the highest peak, while lysine (Lys) shows the lower peak, indicating that the hydrophobic interaction between PEGs and Ala residues is stronger. Glutamic acid (Glu), leucine

Figure 7. Number of clusters (top) and number of the aggregated amphiphilic molecules in the largest cluster (bottom) as a function of time. Snapshots at the beginning and end (1.5 μs) of the simulation, T50-P2S-C18, are shown.

the number of “clusters” and the maximum number of amphiphilic molecules per micelle as a function of time, where a cluster is either a complex of any size or a free amphiphilic trimer. Here, if the distance between carbon-tail beads of different molecules is less than 0.5 nm, then those molecules are considered to be a cluster. These both reach steady states within 600 ns, indicating the equilibration of self-

Figure 6. Radial distribution functions between PEGs and amino acids of the peptide. 8852

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significantly.58 Also, we previously simulated poly(ethylene oxide) (PEO) using the MARTINI FF and calculated Rh from diffusion coefficients, which showed that Rh for large PEOs agree very well with experimental values (small errors within 6%), although the simulated values for small PEOs (Mw < 2000) underestimate experimental values by approximately 20%.46 This indicates that details of the surface become less important in hydrodynamics as the particle size increases. In this work, the self-assembled micelles are much larger than those PEO molecules, and thus a scaling factor of 4, which is typically used for the MARTINI model, is not applied. Figure 8 shows that micelles with more trimers have lower diffusion coefficients, apparently because of the increased size of the micelle. Diffusivities for T50-P2S-C18 and T60-P2S-C18 are very close, indicating that the micelle size is saturated at this concentration of trimer, consistent with Figure 7. For T50-P2SC18 and T60-P2S-C18, over the last 900 ns, these diffusivities of around 0.042 nm2/ns ought to produce a mean-square drift in the position of the micelle of around 38 nm2 or a root-meansquare drift of 6 nm, at which micelles do not interact with each other. In Figure 8, Rh is larger for the micelle with lower diffusivity, as expected. In particular, Rh values for T50-P2S-C18 and T60-P2S-C18 are 6.9 (± 0.6) nm, in quantitative agreement with the calculated outer radius of ∼7 nm as well as the experimentally measured hydrodynamic radius of ∼7.5 nm. These findings indicate that PEGylated amphiphilic trimers self-assemble into a micelle with a radius of ∼7 nm, in quantitative agreement with experiments, while the aggregation numbers of amphiphilic molecules per micelle differ in simulations and experiments. In simulations, the aggregation number of 18 is much lower than the experimental value of 45, presumably because the volume for the headgroup composed of PEGylated helices may be much larger than the volume assumed in the experimentally proposed model. Experimental work should be performed to check these findings and their implications for the effect of the headgroup size on the internal structure of the micelle. Also, all-atom simulation studies of the interactions between PEGs and helices in the micelle ought to be performed, which we hope to report on elsewhere.

assemblies. The number of clusters decreases as the simulation time increases, showing the micelle formation of the randomly distributed amphiphilic trimers, as visualized in Figure 7. The aggregation number of amphiphilic molecules is saturated in systems T50-P2S-C18 and T60-P2S-C18, indicating the aggregation of 18 amphiphilic molecules per micelle. The micelle size was characterized by calculating the outer radii and hydrodynamic radii. The outer radii (R) were calculated from the radial densities of micelles, where the radial density profile (number of beads) is a function of the distance from the COM of the C18 alkyl chains in the micelle core. Figure 8 shows that for T50-P2S-C18 and T60-P2S-C18 the R

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CONCLUSIONS Trimeric coiled coils grafted with PEGs of different sizes and conjugate positions were simulated for 0.8−1 μs. We first performed all-atom simulations of trimeric coiled coils, showing that helices retain their stable α-helical structure with an interhelical distance of ∼1 nm for the whole simulation time, which verifies the CG model for the trimeric coiled coil, where the helical structure is fixed. CG simulations of trimers grafted with PEGs (Mw = 1000, 2000, 5000, and 7000) show that endto-end distances (⟨h2⟩1/2) and radii of gyration (Rg) of PEG chains grafted to the side (Cys, 14th residue) of coiled coils become shorter than those of free PEGs in water, while the size of PEGs grafted to the terminal group of coiled coils is same as the size of free PEGs, in agreement with experiments. Also, the thickness of the PEG layer on coiled coils was calculated from radial densities, showing that the thickness is less than the size of the mushroom, which indicates that the grafted PEGs are more compact than free chains in water. These findings imply the attractive interaction between peptides and PEGs, which is explained by calculating radial distribution functions, showing that hydrophobic residues of peptides tend to be less exposed

Figure 8. Number of micelle beads from COM of the C18 chains in the micelle core (top panel), diffusion coefficients (D) of the COM of the micelle (middle panel), and outer radii (R, calculated from radial densities) and hydrodynamic radii (Rh, calculated from diffusivities) of micelles (bottom panel). The experimental values of R and Rh are from ref 26. Numbers of aggregated amphiphilic molecules are listed in parentheses below the system name.

values are ∼7 nm, close to the experimentally measured radius of ∼7.5 nm.26 This micelle size is also confirmed by calculating hydrodynamic radii (Rh), which can be obtained from diffusion coefficients (D). Diffusion coefficients were calculated from the slopes of the mean-square displacements (MSD) of the COM of the micelle versus time and then correcting these for finite size effects using the formula D = DPBC + kBTξ/6πηL, where kB is Boltzmann’s constant, L is the cubic box length, ξ = 2.837297,57 and the viscosity is η = 0.75 cP at 298 K for CG water.58 Using these diffusion coefficients, we calculated Rh from the Stokes−Einstein equation for a sphere with stick boundary conditions,59 Rh = kBT/6πηD. Note that viscosities of water for the CG model and experiment do not differ 8853

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to water and thus interact with PEGs, which makes PEG chains more compact. In addition, the PEGylated trimer was modified by attaching the C18 alkyl chain and more PEG chains, leading to the amphiphilic trimer. Multiple copies of the amphiphilic trimer, which were initially randomly distributed in the simulation box, self-assemble to micelles because of hydrophobic interactions between C18 chains. The outer radii and the hydrodynamic radii were calculated respectively from radial density profiles and diffusion coefficients, showing that those radii are ∼7 nm, close to the experimental value of 7.5 nm. Although the micelle size is close to the experimentally measured value, the aggregation number of amphiphilic molecules per micelle is 18, which is much lower than the experimentally proposed value of 45, implying that the volume of the headgroup of amphiphilic molecules might be underestimated in experiments. These findings indicate that our simulations predict the experimentally measured size of PEGylated trimers and their self-assembled micelles as well as suggest the lower aggregation number of the micelle, which motivates further systematic experimental studies.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A1001196).

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REFERENCES

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