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J Phys Chem B. Author manuscript; available in PMC 2010 September 13.

Published in final edited form as:

PMCID: PMC2937831

NIHMSID: NIHMS232078

Address reprint requests to Richard W. Pastor, Email: vog.hin.iblhn@rrotsap

The publisher's final edited version of this article is available at J Phys Chem B

See other articles in PMC that cite the published article.

A coarse-grained (CG) model for polyethylene oxide (PEO) and polyethylene glycol (PEG) developed within the framework of the MARTINI CG force field (FF) using the distributions of bonds, angles, and dihedrals from the CHARMM all-atom FF is presented. Densities of neat low molecular weight PEO agree with experiment, and the radius of gyration *R _{g}* = 19.1 Å±0.7 for 76-mers of PEO (

Polyethylene oxide (PEO) and polyethylene glycol (PEG) are polymers with the formulas H_{3}C-O-(CH_{2}-CH_{2}-O)* _{n}*-CH

CG models for PEO have been developed for implicit solvent, and yield good agreement with experiment for chain dimensions,^{24}^{–}^{26} and aggregation number and critical micelle concentration.^{26} Although not computationally demanding, the transferability of implicit solvent models to multicomponent mixtures is limited. Models with explicit solvent are more directly applicable to the interaction of PEG with macromolecular assemblies of lipids and proteins. For example, the CG PEO model with explicit solvent developed by Klein and coworkers showed self-assembly of diblock copolymers in explicit water, and strong interaction with a lipid bilayer.^{27}^{,}^{28} This paper presents a PEO/PEG model suitable for simulations in the MARTINI CG force field (FF) due to Marrink et al.^{29}^{,}^{30} The MARTINI CG FF, originally designed for lipids,^{30} has been extended to proteins^{31} and successfully applied to large assemblies including proteins, nanoparticles, and membranes.^{32}^{–}^{37}

Particles in the MARTINI FF typically consist of 3 or 4 heavy atoms. For PEG/PEO chains, an intuitive building block defines the sequence C-O-C as a CG bead. Then, as illustrated in Figure 1 for *n*=2, an *n*-mer of PEO (denoted PEO*n*) contains *n+1* beads, while an *n*-mer of PEG has *n* beads. In this study, CG molecules are consistently named PEO*n* instead of PEG*n*+1, though can model for either. The parameterization of the bonded interactions between CG beads is primarily based on all-atom simulations of 9, 18, 27, and 36-mers of PEO (442 < molecular weight *M _{w}* < 1,630) in water.

The CG model was tested by comparison to the experimentally determined *R _{g}* of PEG77 (

This paper is organized as follows. Sections 2.1 and 2.2 describe the simulation conditions for the coarse-grained and all-atom simulations, respectively, and Section 2.3 outlines the steps required to evaluate *R _{h}*. The Results and Discussion contains five subsections. Section 3.1 details the parameterization of CG PEO/PEG, and the comparison with all-atom simulations of low molecular weight PEO. Sections 3.2 and 3.3 compare the molecular weight dependence of

Simulations and analyses were performed using the GROMACS simulation package^{47} with the MARTINI CG force field developed by Marrink et al.^{29}^{,}^{30} (downloaded from http://md.chem.rug.nl/~marrink/coarsegrain.html). A cutoff of 12 Å was set for Lennard-Jones (LJ) and electrostatic interactions. The LJ potential was smoothly shifted to zero between 9 and 12 Å, and the Coulomb potential was smoothly shifted to zero between 0 and 12 Å. The pressure was maintained at 1 bar and temperature at 296 K (solutions, and PEO grafted to nonadsorbing surface) or 293 K (neat liquids) by the weak-coupling algorithm.^{48} A time step of 10 fs was used for neat PEO and for the low concentration solutions; an 8 fs time step was employed for the high concentration solutions. Coordinates were saved every 20 ps for analysis. As a final general comment, freezing of pure MARTINI water has been reported at temperatures close to that of the present simulation.^{29} Freezing has been avoided by applying a 21 % radius increase to 10 % of the water particles.^{29} No freezing was observed in any of the PEO/water or PEO/lipid/water simulations carried out for this study, though all waters were identical; i.e., the presence of PEO appears to prevent nucleation. The following subsections provide further details for each system.

Neat PEO1, PEO2, PEO3, PEO4, and PEO5 were simulated to calculate densities. 300 PEO molecules were randomly positioned in an initial periodic box of size 51 Å/side, and simulations were performed for 100 ns, with averages calculated over the last 50 ns.

Single chains of PEO were simulated in water for the following lengths and times: PEO9, PEO18, PEO27, PEO36 (400 ns); PEO44, PEO67 (600 ns); PEO76, PEO90, PEO112, PEO135, PEO158 (800 ns). Four trajectories for each molecular size were generated. To avoid interactions between PEO molecules through the periodic boundary conditions, simulation systems with PEO9-PEO36 included ~9,200 water beads (equivalent to ~36,800 real waters) in box sizes of 100 Å/side; and systems with PEO44-PEO158 included ~16,400 water beads (~65,600 real waters) in 125 Å/side boxes. Analyses were performed with the first 20 ns deleted for PEO9 to PEO36, and the first 100 ns deleted for PEO44 to PEO158.

8 copies of PEO76 were randomly positioned in a box of 130 Å/side (18,544 water beads), leading to a concentration of 21 mg/cm^{3}. Similarly, 72 copies of PEO76 in a 140 Å/side box (19,928 water beads) yielded 148 mg/cm^{3}. These are close to ~30 mg/cm^{3} and ~160 mg/cm^{3}, the concentrations for experimentally measured *R _{g}*.

A model nonadsorbing bilayer was constructed as a mixture of modified DSPC (distearoylphosphatidylcholine) and DSPE-PEO44 (PEO44 attached to the ethanolamine bead in the head group of distearoylphosphatidylethanolamine). Glycerol and head group beads of DSPC and lipid parts of DSPE-PEO44 were changed into hydrophobic C1-type beads which are usually designed for the lipid tail group in the MARTINI FF. The lipid-PEO interaction was also modified to prevent adsorption (LJ: *ε* = 2.0, *σ* = 6.2) without any change of the interactions of PEO-water and PEO-PEO.

Seven systems at grafting densities ranging from 1 to 100% were equilibrated at constant pressure and temperature (NPT), and then simulated at fixed surface area and constant normal pressure (NP_{z}AT). Numbers of DSPE-PEO44 and *total* lipids are as follows: 2, 200; 72, 1820; 98, 1568; 200, 1800; 450, 1800; 900, 1800; 1800, 1800. Each system contained approximately 40 CG waters/lipid, and the surface area ranged from 66 (lowest grafting density) to 80 (highest) Å^{2}/lipid. Simulations were performed for 800 ns for the highest grafting density, and 300 for the others; the last 100 ns were used for analysis.

All-atom simulations and analyses were performed using CHARMM c33b2.^{49} The CHARMM C35r ether force field was used for PEO parameters,^{23}^{,}^{50} and TIP3P model was used for water.^{51}^{,}^{52} Using the velocity Verlet integrator^{53} with a time step of 2 fs, the temperature of 296 K was maintained by the Nose-Hoover thermostat,^{54}^{,}^{55} and the pressure was maintained at 1 atm by the Andersen-Hoover barostat.^{56} Electrostatic interactions were calculated using particle mesh Ewald^{57} and a real space cutoff of 12 Å; van der Waals interactions were switched to zero between 8 Å and 12 Å, and an isotropic long-range correction was applied.^{58} Coordinates were saved every picosecond for analysis.

A single PEO76 was solvated in a box of 74 Å/side (~13,600 water molecules). Two solvated systems were generated with different configurations of PEO76 having initial *R _{g}* of 9 and 24 Å, respectively. Simulations were performed for 70 ns, and the last 50 ns were used for analysis.

5 copies of PEO76 were positioned in a box of 100 Å/side (~35,200 water molecules) for a concentration of 26 mg/cm^{3}; 27 copies of PEO76 were positioned in 100 Å/side (~30,700 water molecules, 146 mg/cm^{3}). These PEO76 concentrations are close to those used in the CG simulation and experiment.^{46} Initial conditions were generated with the *R _{g}* of each PEO ≈ 9, 14 and 27 Å. The experimental

The effective hydrodynamic radius *R _{h}* is a construct obtained from the Stokes-Einstein relationship

$$D=\frac{{k}_{B}T}{f}$$

(1)

where *D* is the translational diffusion constant, *T* is the absolute temperature, *k _{B}* is Boltzmann’s constant, and

$${R}_{h}=\frac{{k}_{B}T}{6\pi \eta D}$$

(2)

*R _{h}* is a useful quantity for comparing models because the simulated diffusion constant

Calculation of *D _{sim}* for each polymer was a multistep process, beginning with

*D _{PBC}* was then corrected for finite size effects using the formula derived by Yeh and Hummer:

$$D={D}_{\mathit{PBC}}+\frac{{k}_{B}T\xi}{6\pi \eta L}$$

(3)

where *L* is the cubic box length, *ξ* = 2.837297, and *η* is the viscosity of the medium. The viscosity increases with higher polymer concentration, and hence the solution viscosity was corrected using the Einstein formula^{45}

$$\eta ={\eta}_{\text{w}}(1+2.5\phi )$$

(4)

where is the volume fraction of the particles, and *η*_{w} is the viscosity of pure water (taken to be 0.75 cP from the value obtained from simulations of the present model at 298 K).^{63} This correction is small, given that ranged from 0.00186 to 0.00346. The final diffusion constant is given by

$${D}_{\mathit{sim}}=\lfloor {D}_{\mathit{PBC}}+\frac{{k}_{B}T\xi}{6\pi \times 0.0075(1+2.5\phi )L}\rfloor $$

(5)

This formula differs from Eq. (6) of ref. ^{23}, where *D _{sim}* was further scaled to the experimental viscosity.

The energy function was designed to be consistent with the MARTINI force field (FF).^{29} In general, the MARTINI potential consists of bond, angle, Lennard-Jones (LJ), electrostatic, and torsional terms. The following functions were parameterized:

$${V}_{\mathit{bond}}(b)=\frac{1}{2}{K}_{b}{(b-{b}_{0})}^{2}$$

(6)

where *K _{b}* is the bond force constant,

$${V}_{\mathit{angle}}(\theta )=\frac{1}{2}{K}_{\theta}{(cos(\theta )-cos({\theta}_{0}))}^{2}$$

(7)

(parameters are defined as for bonds);

$${V}_{\mathit{dihedral}}(\phi )=\sum _{i=1}^{m}{K}_{\phi ,i}(1+cos({n}_{i}\phi -{\phi}_{i}))$$

(8)

where *n _{i}* and

$${V}_{LJ}({r}_{ij})=4{\epsilon}_{ij}\left[{({\sigma}_{ij}/{r}_{ij})}^{12}-{({\sigma}_{ij}/{r}_{ij})}^{6}\right]$$

(9)

where *σ _{ij}* is the zero point of the potential and

Parameters for the CG potential energy function (Table 1) were obtained by matching bond, angle, and dihedral distributions, from all-atom simulations^{23} of PEO9, PEO18, PEO27, and PEO36, and experimental densities of very low molecular weight neat PEO. End-to-end distributions and *R _{g}* were used to evaluate the parameters. The following paragraphs describe the critical considerations for refining each of the terms.

In the MARTINI FF, values of *σ _{ij}* (in Å) are set to 4.7 for particle types representing approximately 4 heavy atoms, and 4.3 for smaller particles representing 2–3 atoms including those in rings. For particles with

Parameters for the bond, angle and dihedral potentials were obtained by comparing distributions from our previous all-atom simulations of PEO36,^{23} after mapping them to the CG representation. Following the conventions for development of the MARTINI FF, distributions were evaluated from the centers of mass for each monomer (C-O-C) in the all-atom model, leading to unimodal distributions for bonds and angles, and a bimodal one for dihedrals (Figure 2, dashed lines). The solid lines in Figure 2, obtained from simulations of PEO36 based on the CG parameters listed in Table 1, agree very well with the all-atom set, demonstrating the success of the parameterization of the bond and angle terms. The standard MARTINI approach does not include proper torsional terms. CG dihedral distributions for alkanes and alkenes can be matched to atomistic distributions sufficiently well using bond and angle potentials in combination with the nonbonded terms that are excluded only between nearest neighbors. For peptides and proteins, elastic bonds have been used to match the specific dihedral distributions for secondary structure elements (extended versus helical arrangements). The preceding strategies failed here, and explicit proper torsional terms were required. Figure 2 shows the very good correspondence of CG and the mapped atomistic dihedral distributions. A similar result for a different CG model of PEO was reported by Fischer et al.^{25}

Probability distributions of bond lengths (*b*, top), angles (*θ*, middle), and dihedrals (, bottom) of PEO36 from atomic^{23} and CG models.

Nonbonded parameters were optimized on the basis of comparison of densities of low molecular weight PEO to experimental values as well as comparison of the radius of gyration and end-to-end distance distributions of single PEO chains in water to mapped atomistic simulation results. Table 2 compares simulated and experimental densities of low molecular weight PEO. Differences of <5% in PEO3 to PEO5 are comparable to those obtained in the parameterization of CG alkanes.^{30}

Simulated and experimental^{72} densities of low molecular weight PEO*n*. PEO1 corresponds to dimethoxyethane, CH_{3}-O-CH_{2}-CH_{2}-O-CH_{3} and PEO2 is CH_{3}-O-CH_{2}-CH_{2}-O-CH_{2}-CH_{2}-O-CH_{3}.

Table 3 lists *R _{g}* and

Radius of gyration *R*_{g} and root mean squared end-to-end distance *h*^{2}^{1/2} for coarse-grained and all-atom PEO of length *n*. The value at each molecular weight (*M*_{w}) is averaged from the four and two replicate simulations, respectively for **...**

$$\mathit{Prob}\phantom{\rule{0.16667em}{0ex}}(h)=\frac{4\pi {h}^{2}{C}_{1}}{L{(1-{(h/L)}^{2})}^{9/2}}exp\left(\frac{-3L}{4\lambda (1-{(h/L)}^{2})}\right)$$

(10)

where *L* is the length of the fully extended (all-atom) polymer.
${C}_{1}={({\pi}^{3/2}{e}^{-\alpha}{\alpha}^{-3/2}(1+3{\alpha}^{-1}+\frac{15}{4}{\alpha}^{-2}))}^{-1}$, *λ* is the persistence length, and *α* = 3*L*/4*λ*. Agreement of the simulated and analytic distributions is very good for the 4 lengths shown for *λ* = 3.7 Å (the value of the all-atom simulations and experiment).

A minor peak at approximately 5 Å (close to *σ _{ij}*) is evident for PEO9 (Fig. 3). This implies that a small subset of the lower molecular weight PEO forms ring-like conformations not observed in the MD simulations or in the worm-like chain model. The peak gradually diminishes by PEO36. Further refinement of the CG model to include a separate bead type for the terminal groups could likely eliminate this conformation. However, given that this artifact is restricted to the low

Figure 4 plots the instantaneous *R _{g}* and

The present range of PEO allows an examination of the transition from ideal to real chain behavior; i.e., a determination of where the coefficient *ν* in
${R}_{g}\propto {M}_{w}^{\nu}$ shifts from *ν* = 0.5 to the experimental value of 0.583. Figure 5 plots log *R _{g}* vs. log

Columns 2 and 3 of Table 4 list *D _{PBC}* and

At low polymer concentration, excluded-volume effects are predominantly intramolecular, and cause the coil to swell. As concentration increases, intermolecular excluded-volume interactions reduce the size of each coil.^{46}^{,}^{65} The cross-over point for PEO, however, is not well established. Small angle neutron scattering (SANS) measurments^{46} on PEG77 indicate a decrease in the *apparent R _{g}* from 19.7 Å at 30 mg/cm

Figure 6 shows the average *R _{g}* of the sets of PEO76 for the all-atom simulations at low and high concentrations (see Section 2.2.b for nomenclature). The average

Average radii of gyration of five all-atom PEO76 at concentration of 26 mg/cm^{3} (L27, L14, L9), and 27 all-atom PEO76 at 146 mg/cm^{3} (H27, H14, H9). (See Section 2.2.b for nomenclature)

To increase solubility and circulating time of drug molecules, PEG has been attached to the drug transporters such as vesicles, micelles, and nanoparticles,^{2} which has motivated theoretical and experimental studies of the behavior of PEG grafted on various surfaces.^{67}^{–}^{71} The theoretical treatment by Alexander^{21} and de Gennes^{22} defines two regimes for a polymer grafted on a nonadsorbing surface. At very low grafting density, the chain behaves much like an isolated chain in solution (with the obvious proviso that half of the conformational space is excluded by the surface). Consequently it traces out a hemisphere (or “mushroom”) with a size given by the Flory radius

$${R}_{F}\equiv {aN}^{3/5}$$

(11)

where *N* is the degree of polymerization and *a* is the monomer size; for this study, *a* is set equal to the bond length *b _{0}* = 3.3 Å.

$$L=Na{(a/D)}^{2/3}$$

(12)

where *D* is the distance between the grafting points of polymers in the lateral plane. The mushroom and brush states are sketched on the top row of Figure 7. This subsection applies Alexander-de Gennes theory to the results of simulations of PEO44 (N=45) grafted to a bilayer modified to model a hydrophobic nonadsorbing surface. Of particular interest is a consistent specification of *L* that can be applied to simulations of PEG on more complicated surfaces.

Schematic representations of the mushroom and brush configurations (top), and snapshots of the side (middle) and top-down (bottom) views at the end of simulations of PEO44 (red) and a hydrophobic surface (blue) with *D* = 44 Å (left) and *D* = 12 **...**

As described in Section 2.1.d, lipid bilayers consisting a mixtures of modified DSPC and DSPE-PEO44 were simulated; Table 6 lists the mole fractions of DSPE-PEO44 and *D*-values for the 7 systems. The middle and bottom rows of Figure 7 show the side and top-down views, respectively, of the final conformations of the grafted PEO44. It is clear that the polymer is in mushroom and brush-like states at *D*=44 and 12 Å, respectively. Figure 8 plots the densities of PEO44 normal to the surface for *D*=9 to 81 Å. The distributions are reasonably similar at high *D*, and become more extended as *D* decreases and the chains overlap.

Density probabilities of PEO44 as a function of distance from the PEO44-surface interface (*D* = 9–81 Å).

Size of mushroom and thickness of brush calculated from simulations of PEO44 (*N* = 45, *a* = 3.3 Å) and the Alexander-de Gennes theory.^{22} Values from simulations were averaged over the last 100 ns. All lengths are in Å.

Table 6 lists <*h*^{2}>^{1/2} for each value of *D*. For *D* =35–81 Å, <*h*^{2}>^{1/2} = 28–33 Å; i.e., nearly equal to *R _{F}* = 32 Å. For

A coarse-grained (CG) model for polyethylene oxide (PEO) and polyethylene glycol (PEG) was developed within the framework of the MARTINI CG force field. Densities of neat PEO1 to PEO5, and distributions of bond, angle, dihedral, and end-to-end distances of PEO9 to PEO36 in water are in excellent agreement with those from experiments and all-atom simulations. Simulations of PEO9 to PEO158 (442 < *M _{w}* < 6998) in explicit water then yielded 1600 <

The hydrodynamic radii *R _{h}* for the longer chains (>PEO67) are in very good agreement with those from experiment; the lower

Average *R _{g}* calculated from CG simulations of multiple copies of PEO76 are 18.9 ± 1.1 Å at low concentration (21 mg/cm

The effective lengths (set to 97–99% of the density above the surface) of PEO44 grafted on hydrophobic (nonadsorbing) surface are described well by Alexander-de Gennes theory. The transition between mushroom and brush occurs at *D* = 27–35 Å, where *D* becomes close to the Flory radius *R*_{F}=32 Å for this length polymer. The agreement of theory and simulation, after an effective brush thickness is defined, provides the basis for analysis of more complex systems including PEGylated bilayers. However, the PEO-lipid interaction parameters must be developed before the present model can be applied to such systems in the MARTINI FF.

We thank Richard Venable, Edward O’Brien, Monica Pate, and P. Thiyagarajan for helpful discussions. This research was supported by the Intramural Research Program of the NIH, National Heart, Lung and Blood Institute. This study utilized the high-performance computational capabilities of the CIT Biowulf/LoBoS3 and NHLBI LOBOS clusters at the National Institutes of Health, Bethesda, MD.

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