Coil–Globule–Coil Transition of PNIPAm in Aqueous Methanol

Article pubs.acs.org/Macromolecules

Coil−Globule−Coil Transition of PNIPAm in Aqueous Methanol: Coupling All-Atom Simulations to Semi-Grand Canonical CoarseGrained Reservoir Debashish Mukherji and Kurt Kremer Max-Planck Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany S Supporting Information *

ABSTRACT: Conformational transitions of (bio)macromolecules in aqueous mixtures are intimately linked to local concentration fluctuations of different solvent components. Though computer simulations are ideally suited to investigate such phenomena, in conventional setups the excess of one cosolvent close to the solute leads to depletion elsewhere, requiring very large simulation domains to avoid system size effects. We, here, propose an approach to overcome this depletion effect, which combines the adaptive resolution scheme (AdResS) with a Metropolis particle exchange criterion. In AdResS, a small all-atom region, containing the solute, is coupled to a coarse-grained reservoir, where the particle exchange is performed. The particle exchange would be almost impossible had they been performed in an all-atom setup of a dense molecular liquid. As a first application of the method, we study the concentration driven reentrant collapse and swelling transition of poly(N-isopropylacrylamide) (PNIPAm) in aqueous methanol and demonstrate the role of the delicate interplay of the different intermolecular interactions. to atom overlap.13 If, instead, the molecules are represented by spherically symmetric coarse-grained (CG) beads, the insertion of another bead is typically much easier. To overcome these problems, an efficient open boundary scheme for complex molecular liquids, which allows for the insertion of molecules in dense incompressible fluids, will be proposed in this work. A closer look into the macromolecular solvation suggests that the specific chemical details are only important within the correlation length originating from the solute structure. The rest of the bulk solution is needed to maintain solvent equilibrium, and consequently does not require chemical details and can safely be represented by single site CG beads. For this purpose, a scheme is needed that can couple the chemically specific AA region to a CG reservoir, provided that there is no free energy barrier in the coupling and the structure and thermodynamics are preserved, at-least, within the AA region. One such method is the adaptive resolution molecular dynamics scheme (AdResS).14,15 The AdResS scheme is based on force interpolation in the transition zone from CG to AA.14 More recently, a Hamiltonian based scheme (HAdResS), where interaction energies are interpolated, has been proposed.16 Here we will use the AdResS scheme based on force interpolation. It is however straightforward to employ the very same procedure within the H-AdResS setup. Our method makes use of AdResS scheme in conjunction with a Metropolis particle exchange criterion and is, thus, suited to study systems with mixed solvents. The particle exchange is performed within

1. INTRODUCTION Establishing the links between microscopic (bio)macromolecular structure and their physical properties is a key to our understanding of biological systems. (Bio)molecular solvation is one such case, where the interaction of dissolved macromolecule with water and cosolvents dictates whether a macromolecule remains in a folded globular conformation or extends into a coil.1−3 While there are many experimental investigations of macromolecular solvation in aqueous mixtures,3−6 their molecular level understanding, often, still is a matter of debate.7−10 The better theoretical understanding of such complex molecular phenomena would require: (a) information on (bio)physical processes occurring over long time scales and (b) large simulation domains to “reasonably” mimic experimental conditions. The latter become much more challenging when the conformational transitions are driven by large local solvent concentration fluctuations.7,10,11 For such problems, conventional all-atom (AA) simulations of midsized closed boundary domains do not maintain solvent equilibrium with the bulk solution. Moreover, the solvent equilibrium can somehow be maintained by choosing enormously large simulation domains,10,12 where attaining long time scales, for the equilibrium conformational sampling of large macromolecules, can thus be computationally very expensive, especially when the full chemical details are included. Therefore, the best possible approach would be to employ a grand canonical simulation, where the solvent equilibrium can be maintained by inserting and/or deleting solvent molecules within the simulation domain. However, when dealing with AA representations of complex molecular solvents, the insertion of molecules is usually hindered by very poor acceptance rates due © XXXX American Chemical Society

Received: September 11, 2013 Revised: October 15, 2013

A

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⎡ ⟨NN δij ⎤ i j⟩ − ⟨Ni⟩⟨Nj⟩ ⎥ − Gij = V ⎢ ⟨Ni⟩⟨Nj⟩ ⟨Nj⟩ ⎥⎦ ⎢⎣

the CG region of the AdResS setup. The method produces well converged solvation properties using a small simulation domain, which otherwise would not be possible within conventional molecular dynamics of similar system sizes without particle exchange. We apply this approach to poly(Nisopropylacrylamide) (PNIPAm) in aqueous methanol and show how the experimentally observed coil−globule−coil transition1 is connected to the variation of the methanol excess coordination around a NIPAm monomer. The remainder of the paper is organized as follows: In section 2, we sketch the method where we give a brief summary of AdResS method together with Kirkwood-Buff theory of solutions. The results are presented in section 3 and finally we give our conclusions in section 4.

= 4π

∞

[gijμ VT(r ) − 1]r 2 dr

(2)

where averages are denoted by brackets ⟨·⟩. V is the volume, Ni is the number of particles of species i, δij is the Kronecker delta, gμVT ij (r) is the radial distribution function in the μVT ensemble. When a single solute (s) is present in a mixture of water (w) and cosolvent (c), the change in the chemical potential of the solute μs is given by21 1 ⎛ ∂μs ⎞ ⎜⎜ ⎟⎟ RT ⎝ ∂ρc ⎠

2. METHODOLOGY

= p,T

Gsw − Gsc 1 − ρc (Gcw − Gcc)

(3)

where ρc is the cosolvent number density and R is the gas constant. Ideally Gij should be calculated at r → ∞.22 Practically, a cutoff rc has to be introduced where gμVT ij (r) is indistinguishable from unity. For the present study this results in rc between 0.9 and 1.3 nm.

2.1. The Adaptive Resolution Molecular Dynamics Scheme. The AdResS14,15 is a multiscale scheme, where an allatom region can be coupled to a CG reservoir and particles are allowed to change their spatial resolution during the simulation. The transition is governed by a weighted function w(r), which is unity for the explicit system, zero for the coarse-grained, and smoothly varies between zero and unity in the hybrid region. This coupling uses interpolated forces between molecules α and β, cg Fαβ = w(rα)w(rβ)Fexp αβ + [1 − w(rα)w(rβ )]F αβ

∫0

3. RESULTS AND DISCUSSIONS The AA simulations are performed using GROMACS molecular dynamics package.23 We use the Gromos96 force field24 for methanol, the SPC/E water model25 and the force field parameters for PNIPAm are taken from ref 26. Unless stated otherwise, the temperature is set to 298 K. We perform an initial volume equilibration of the all-atom reference system using a Berendsen barostat27 at 1 atm pressure using a coupling time of 0.5 ps for 1 ns. The AdResS simulations are performed using a modified GROMACS code.28 The nonbonded interactions in the CG region of AdResS is described by a single site model derived using the iterative Boltzmann inversion (IBI).29 Because of the nontransferability of structure based CG potentials, we have derived nonbonded potentials for each concentrations separately. The electrostatics are treated using particle mesh ewald in AA simulations and reaction field is used for the AdResS setup. The reaction field dielectric constant εr is calculated from the AA simulation trajectory.12 System sizes used for the simulations are described in the Supporting Information. 3.1. Kirkwood−Buff Integrals. In Figure 1, we show water−water KBI (Gww) in aqueous methanol mixture at a methanol mole fraction xm = 0.90. Looking at the AA data for two different system sizes (see the green and the black curves in Figure 1), it is apparent that an approximate convergence is observed only for the large system of 21000 molecules. This is NVT not surprising, since the approximation gμVT ij (r) ≈ gij (r) only holds for large systems. Furthermore, the nonconvergence of KBI, based on AA data of 6000 molecules, is a clear indication of the solvent depletion in a small-sized simulation domain. We would also like to point out that we have only chosen two system sizes because approximately a 21000 molecules system can give reasonable convergence in Gww(r) (see for example Figure 2 in ref 12). For comparison, we have also performed conventional AdResS simulations14,15 using the approach presented in earlier works.11,12 A typical AdResS setup for the aqueous methanol mixture of 21000 molecule system is shown in part a of the Figure 2 (for details see the Supporting Information). The Gij(r)’s are calculated within the AA region of the AdResS setup. It can be appreciated in the Figure 1 that Gww(r)

(1)

Fαβ is the total intermolecular force acting between two molecules and Fexp αβ is the sum of all high resolution pairwise cg interactions between atoms of molecules α and β. Fcg αβ = −∇Vαβ cg is the pairwise coarse-grained force based on Vαβ, the pairwise coarse-grained potential. rα and rβ are the distances of the molecular centers-of-mass from the center of the simulation domain. The molecules change their spatial resolutions (from all atom to coarse-grained and vice versa) on-the-fly, allowing for free exchange of particles. When studying the solvation of macromolecules, the solute is always constrained at the center of the explicit region. The virial pressure of the CG model is higher than the reference all-atom system.17 Therefore, the AdResS coupling can lead to unphysical particle drift from the CG region to the AA region, leading to the unrealistic solvent composition in the explicit region. On the other hand, if we use pressure correction,17 the compressibility of the CG model will be different. Therefore, it turns out to be a trade-off between the correct compressibility or the correct pressure. In our case, we prefer to preserve compressibility and, therefore, do not employ any pressure correction for the derivation of the CG nonbonded interactions. Moreover, the nonuniform density profile can be healed by using an iterative thermodynamic force for the individual solvent components,18,19 which is dependent on the slopes of the density profiles within the hybrid region. A detail is mentioned in section II of the Supporting Information. 2.2. Kirkwood-Buff Theory of Solutions. We are interested to study the solvation thermodynamics of solutions. Therefore, we start our discussion with the fluctuation theory of Kirkwood and Buff (KB),20 which connects the density fluctuations in the grand canonical ensemble (μVT) to the integral of the radial distribution function via KB integrals (KBI), B

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reasonable result. This would require further increase of the solvent box when a macromolecule is added. The larger the solute the larger a solvent box is needed to avoid severe depletion effects. Therefore, it is worth investigating whether we can resolve the depletion within the small system of 6000 molecules, by a grand canonical type approach. 3.2. Semi-Grand Canonical Scheme: Particle Exchange AdResS. Open systems molecular dynamics (MD) is of particular importance for the simulation of complex biomolecular systems. Previously, it was shown that the coupling of the AdResS scheme with the continuum description can enable the possibility of truly open systems MD.30 Here, however, we take a much simpler route. When dealing with laboratory experiments, it is evident that irrespective of the local concentration fluctuations the bulk solution is always kept at a constant density to a very good approximation. Mimicking this condition in molecular simulations require either an enormously large solvent bath, such that local aggregation of one species does not induce sizable depletion away from the solute, or a particle insertion criterion within a small-sized simulation domain that can maintain solvent equilibrium. In this work, we devise a protocol consistent with the latter case. In part b of Figure 2, we show the schematic representation of the open boundary setup. Since the size of the AA region and the hybrid region in part b of Figure 2 are kept the same as part a of Figure 2, the size of the CG reservoir is reduced significantly. For example, in the conventional AdResS scheme the average number of particles in the CG region is ≈19000, which is now reduced to only ≈4000 within the new setup. To employ the particle exchange criterion, we use an identity swap protocol suggested for polymeric systems.31 In this protocol, the total number of particles within the simulation domain remain constant. However, the numbers of individual components are allowed to fluctuate. For this purpose, we monitor the density of individual components of aqueous methanol mixture within the CG region after every 50 ps of AdResS run. If depletion is observed, then the identity of particles is swapped based on a standard Metropolis criterion. The particle exchange is performed using a script32 compatible with the modified GROMACS AdResS code.28 We find typical acceptance rates of these exchange moves of approximately 80% when a methanol is replaced with a water molecule and 3% otherwise. We want to emphasize that the particle exchange, at such high acceptance rates, is only possible because the replacements are done within the CG region, where the molecules are represented by spherically symmetric single site beads. These exchange moves would have severely suffered from poor acceptance rates, had they been performed in a full AA setup. Furthermore, we perform the particle replacements at the eight corners of the simulation domain (see part b of Figure 2), i.e., the farthest from the chemically specific center and beyond the interaction range with molecules in the hybrid region. This eliminates the possibility of any residual effect on the AA domain that might occur due to the particle exchange. For the remainder of the paper we will refer to this hybrid method as particle exchange AdResS or PE-AdResS. Result for the PE-AdResS is shown in Figure 1 (see brown curve). Indeed, PE-AdResS gives a well converged Gww(r) for very small systems, supporting the robustness of our approach. It is yet important to mention that even when the PEAdResS reproduces well converged KBIs, the method is not “only” restricted to the precise calculation of Gij’s. Earlier studies33,34 have shown that the well converged Gij’s can be

Figure 1. Kirkwood−Buff integrals (KBI) of aqueous methanol mixtures calculated between water molecules Gww for methanol mole fraction of xm = 0.90. Results are shown for three different methods and two different system sizes. Note: we chose this extreme concentration where the system size effects are most severe. Once we ensure a convergence of KBI at this given concentrations, the other concentrations will always show proper convergence. Temperature is set to 298 K.

Figure 2. Schematic representation of the schemes used for the simulations: (a) Conventional adaptive resolution scheme (AdResS), where a small AA region is coupled to a large “closed boundary” coarse-grained reservoir. (b) Particle exchange adaptive resolution scheme (PE-AdResS), where an AA region is coupled to a much smaller open boundary coarse-grained reservoir, where particle exchange is performed at the eight corners of the simulation domain to avoid depletion effects. Note that only four corners are shown because of the two-dimensional representation of the simulation domain. In both simulation setups, the size of the AA is chosen as 1.4 nm and the width of the hybrid region, which is located between the CG and AA regions, is chosen as 1.2 nm. (c) Mapping scheme representing the smooth coupling between AA and CG particle representations. In the AA description: oxygen is shown in red, hydrogen in silver, and the united atom CH3 is shown in steel. Coarsegrained representation: the red and silver beads represent methanol and water, respectively.

obtained from AdResS simulation matches reasonably well with the AA data of the same overall system sizes. Furthermore, it is perceptible that, already for the binary methanol−water mixture, a large simulation domain is needed to get any C

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NIPAm−methanol Gpm and NIPAm−water Gpw KBIs with varying xm. As expected, PE-AdResS results from small systems give perfect agreement with the AA data from the much larger simulation setup. Furthermore, two important features are seen from Figure 3: (1) Gpm shows an excess coordination between the concentration range 0.1< xm < 0.4 and (2) Gpw first decreases with increasing xm and then starts to increase for xm > 0.4. More interestingly, this concentration range is in reasonably good agreement with the experimental range (0.1 < xm < 0.5) over which the LCST is reduced.38 It also supports an analysis based on a Flory−Huggins theory, suggesting that the water−methanol coordination is preferred over PNIPAmwater hydrogen bonds38 in the concentration regime of the globular PNIPAm state. This observation is clearly visible in our simulations, where the “salting-in” (Gpm > Gpw) inhibits PNIPAm-water coordination, thus is responsible for the reduced capability of hydrogen bonding between PNIPAm and water, even when the water population is in majority. While PE-AdResS simulation of a NIPAm gives reasonably consistent results comparable to the experimentally observed LCST data38 and thus the microscopic insight into the coil− globule−coil transition of PNIPAm in aqueous methanol, it would still be interesting to directly investigate the collapse transition of PNIPAm.26 Therefore, as a first application and final test of our PE-AdResS approach, we simulate a single chain of PNIPAm in aqueous methanol mixture for two different chain lengths Nl = 20 and Nl = 40. When simulating a large PNIPAm, the excess of methanol near the polymer (as observed in Gpm) would severely deplete methanol molecules away from the solute structure in conventional simulation setups, leading to uncontrollable simulation artifacts. The PEAdResS instead mimics the experimental condition closely and allows for a more reasonable comparison with experimental results. In Figure 4, we show radius of gyration Rg as a function of xm (see Supporting Information for simulation snapshots). The data clearly support the collapse and swelling transition of PNIPAm. Furthermore, the collapsed transition is associated with the Gpm excess. To even better illustrate the comparison between experiment and simulation, we have plotted the inset in Figure 4. Given that we are studying complex macromolecular folding, the concentration range over which the polymer collapses is found to be in a very good, almost quantitative, agreement. Lastly, we focus on the thermodynamic analysis, which can easily be computed using KB theory. In Figure 5, we show

obtained by mathematically modifying gij(r)’s. While these methods are suitable to investigate thermodynamic properties of binary and/or ternary mixtures, they do not heal the problem of depletion far from the chemically specific center. Therefore, to further demonstrate the robustness of our method, we study the conformational transition of PNIPAm in aqueous methanol mixtures. 3.3. PNIPAm Solvation in Aqueous Methanol. The study of PNIPAm is of particular interest because it shows an inverse coil−globule transition in pure water; i.e., it remains in the coil state at low temperature and collapses into a compact globule upon increase of temperature. The solution has a lower critical solution temperature (LCST) of about 305 K, which has been studied both experimentally35 and theoretically.36,37 Moreover, when adding methanol to water, the LCST first decreases with increasing xm and then increases for xm > 0.5.38 As a result, PNIPAm shows an interesting reversible coil− globule−coil transition over full methanol concentration range xm1,26 at ambient temperatures. The origin of the large scale conformational collapse stems from the microscopic solute− solvent preferential interactions. Therefore, we first show that our method gives accurate data about preferential interactions within the framework of KB theory. In Figure 3, we show

Figure 3. Kirkwood−Buff integrals (KBI) between solute−solvent pairs, monomer-methanol Gpm and monomer-water Gpw, as a function of methanol mole fractions xm. KBIs are calculated by averaging Gij(r) between 0.9 and 1.3 nm. Temperature is set to 298 K. Continuous curves are guides to the eye.

Figure 4. Radius of gyration Rg as a function of methanol mole fraction xm. In part a, we have also included the polynomial fitted curves from Figure 3 to emphasize that the increased methanol coordinations around NIPAm is consistent with the collapsed transition of PNIPAm. Data is shown for two different chain lengths Nl. In part b, we have shown PE-AdResS simulation data for Nl = 40, together with the experimental data of degree of swelling q taken from ref 26. The data is shown for PE-AdResS simulation and the temperature is set to 298 K. Arrows are pointing at the corresponding y-axis of a particular data set. D

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AdResS scheme is of particular importance because it enables us to easily couple an AA region with a CG reservoir, where the particle exchange is performed. The particle exchange would be (almost) impossible in an AA setup of a dense incompressible fluid. Here we focus on the solvation properties using the Kirkwood−Buff theory of solutions. Our method produces well converged Kirkwood-Buff integrals that are very difficult to obtain within brute force AA MD of even larger system sizes without particle exchange. This is illustrated by the coil− globule−coil transition of a poly(N-isopropylacrylamide) (PNIPAm) in aqueous methanol mixture. While we have shown examples of aqueous methanol mixtures and solvation of PNIPAm in aqueous methanol mixtures, the application of this methods is not at all restricted to biomolecular simulations. More specifically, the approach also has its potential applicability in studying any system of dense fluids that need a grand canonical type simulation environment, such as crystallization from mixtures near adhering surfaces.

Figure 5. Chemical potential shift per monomer μp/Nl as a function of methanol mole fraction xm for two different chain lengths Nl. Data for single monomer are also included. However, the μp of a NIPAm is scaled by a factor of 239 to obtain the master curve. Line is a linear fit to the data within the range 0.1 ≤ xm ≤ 0.5. The μp is calculated by integrating the data obtained from eq 3. Temperature is set to 298 K. We also show simulation snapshots for Nl = 40 showing extended coil for pure water and for xm = 0.750 and collapsed globule for xm = 0.100. Solvent molecules are drawn only within 0.7 nm from the solute structure.

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ASSOCIATED CONTENT

S Supporting Information *

Simulation details, thermodynamic forces in AdResS, KBIs in pure solvent, and simulation snapshots. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

chemical potential shift of PNIPAm μp as a function of xm for different Nl’s, calculated by integrating eq 3. The data corresponding to Nl = 1 is scaled by a factor of 2.39 It can be appreciated that the data sets for different Nl fall on a master curve. Furthermore, μp first decreases with increasing xm even though the solvent quality remains poor, reaching a reasonable plateau for xm ≥ 0.5. While the data clearly suggests a preferential solubility for xm ≥ 0.5, it still needs to be understood why we observe a gradual decrease within μp for xm < 0.5 even when the chain remains in a globular conformation. This counterintuitive behavior can be understood from the simulation snapshots, where it has become apparent that even when Gpm > 0 for the range 0.1 ≤ xm < 0.4 there are a few methanol molecules to stabilize the extended coil structure of the PNIPAm via hydrogen bonding, whereas NIPAm−water coordination suffers from depletion. Moreover, with the increase of methanol concentrations, PNIPAm and methanol hydrogen bonding gets dominant and again results in the expansion of PNIPAm for xm ≥ 0.5 (see the simulation snapshots in Figure 5). Therefore, our analysis suggests that the reentrant coil−globule−coil transition is a result of the delicate interplay between μp and the hydrogen bonding network that originate because of the preferential solute−solvent interactions. To get a quantitative estimate, we have also calculated average number of hydrogen bonds between PNIPAm and solvent molecules. Within the concentration range where polymer collapses into globule, we observe ≈20% decrease in the average number of hydrogen bonds.

*E-mail: (D.M.) [email protected]. *E-mail: (K.K.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.M. thanks Francisco Rodriguez-Ropero for providing the PNIPAm force field. The development of this work has greatly benefited from the fruitful collaborations with Nico van der Vegt and Luigi Delle Site, which we take this oppurtuinity to gratefully acknowledge. We thank Jia-Wei Shen, Raffaello Potestio, and Jalal Sarabadani for critical reading of the manuscript. Snapshots in this manuscript are rendered using VMD.40

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4. CONCLUSIONS We have presented an efficient and versatile method to study solvation thermodynamics of the macromolecules in aqueous mixtures. By combining the adaptive resolution scheme (AdResS) with a Metropolis particle exchange criterion, we propose an open boundary method that heals the particle depletion within a small-sized closed boundary setup. The E

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